CTJan27 Online Year 9 - Probability

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Foundations of Probability

This lesson introduces the fundamental concepts of probability, which is the measure of the likelihood that an event will occur.

1. Sample Spaces and Events

  • Sample Space (S): The set of all possible outcomes of an experiment. Example: When rolling a standard six-sided die, the sample space is $S = \{1, 2, 3, 4, 5, 6\}$.

  • Event (E): A subset of the sample space. It is a collection of one or more outcomes.

    • Simple Event: An event with a single outcome. Example: Rolling a 3. The event is $E = \{3\}$.
    • Compound Event: An event with more than one outcome. Example: Rolling an even number. The event is $E = \{2, 4, 6\}$.

2. Theoretical Probability

When all outcomes in a sample space are equally likely, the theoretical probability of an event E is calculated using the formula:

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{n(E)}{n(S)}$

Example: The probability of rolling an even number on a fair six-sided die.

  • The number of favorable outcomes (even numbers) is $n(E) = 3$ (the numbers 2, 4, 6).
  • The total number of outcomes is $n(S) = 6$.
  • Therefore, $P(\text{even}) = \frac{3}{6} = \frac{1}{2}$.

3. Complementary Events

The complement of an event E, denoted as $E'$ or $E^c$, consists of all outcomes in the sample space that are not in E. The Complement Rule states:

$P(E') = 1 - P(E)$

Example: What is the probability of not rolling a 5 on a fair die?

  • First, find the probability of rolling a 5: $P(5) = \frac{1}{6}$.
  • Using the complement rule, the probability of not rolling a 5 is: $P(\text{not } 5) = 1 - P(5) = 1 - \frac{1}{6} = \frac{5}{6}$.

4. Types of Probability

  • Theoretical Probability: Based on mathematical reasoning and the assumption of equally likely outcomes. This is what we've calculated so far.

  • Experimental Probability: Based on the results of an actual experiment or observation. It is an estimate of the theoretical probability.

$P(E) \approx \frac{\text{Frequency of event E}}{\text{Total number of trials}}$

Example: If you flip a coin 100 times and it lands on heads 54 times, the experimental probability of getting heads is $\frac{54}{100} = 0.54$.

  • Subjective Probability: Based on personal judgment, experience, or intuition. It's an educated guess. Example: A doctor estimating a patient has a 70% chance of full recovery.

5. Simulation

A simulation is a process that models a real-world situation to estimate probabilities. It's a way to conduct a large number of trials to find an experimental probability, especially when the theoretical calculation is complex or impossible.

Example: To estimate the probability of a family with 4 children having exactly 2 boys, you could flip a coin 4 times (letting heads=boy, tails=girl) and repeat this process many times, recording the results.

1

A spinner is divided into 4 equal sections, colored Red, Green, Blue, and Yellow. An experiment is conducted where the spinner is spun 80 times. The color Green came up 25 times. Based on this information, what is the theoretical probability of landing on Green in a single spin?

$P(G) = \frac{3}{4}$

$P(G) = 1 - \frac{25}{80} = \frac{11}{16}$

$P(G) = \frac{25}{80} = \frac{5}{16}$

$P(G) = \frac{1}{4}$
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Defining and listing sample spaces

Questions for: Defining and listing sample spaces

2

What is the sample space, $S$, for rolling a single standard six-sided die?

$S = \{\text{Even}, \text{Odd}\}$

$S = \{0, 1, 2, 3, 4, 5\}$

$S = \{1, 2, 3, 4, 5, 6\}$

$S = \{1, 6\}$
3

An experiment consists of tossing two fair coins simultaneously. Which of the following represents the correct sample space for this experiment? (Let H = Heads, T = Tails)

$S = \{HT, TH\}$

$S = \{HH, HT, TH, TT\}$

$S = \{HH, TT\}$

$S = \{H, T\}$
4

A coin is tossed once and a standard six-sided die is rolled once. What is the total number of possible outcomes in the sample space?

2

12

6

8
5

A bag contains 3 red marbles, 2 blue marbles, and 5 green marbles. One marble is drawn at random. What is the sample space for the color of the marble drawn?

$S = \{\text{Red}, \text{Blue}, \text{Green}\}$

$S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$

$S = \{R, R, R, B, B, G, G, G, G, G\}$

$S = \{3, 2, 5\}$
6

A student is chosen at random and asked to name their favorite day of the school week (defined as Monday to Friday). Which of the following is the sample space for their response?

$S = \{\text{School, Home}\}$

$S = \{\text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}\}$

$S = \{\text{Monday, Tuesday, Wednesday, Thursday, Friday}\}$

$S = \{\text{Weekday, Weekend}\}$
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Identifying simple and compound events

Questions for: Identifying simple and compound events

7

An experiment consists of rolling a standard six-sided die. Which of the following is a simple event?

Rolling a number greater than 2

Rolling an even number

Rolling a prime number

Rolling a 5
8

From a standard 52-card deck, one card is drawn. Which of the following is a compound event?

Drawing a King

Drawing the Jack of Hearts

Drawing the Ace of Spades

Drawing the 10 of Clubs
9

An experiment involves flipping a coin and then rolling a six-sided die. The outcomes are pairs, like (Heads, 3). Which of the following events is a simple event for this experiment?

Getting an even number on the die

Getting a tail on the coin and a 4 on the die

Getting a head on the coin and a number less than 5 on the die

Getting a head on the coin
10

A bag contains 3 red marbles, 2 blue marbles, and 5 green marbles. Which of the following describes a compound event when drawing one marble?

Drawing a specific marble marked with an 'X'

Drawing a red marble

Drawing the third green marble that was placed in the bag

Drawing the first blue marble that was placed in the bag
11

Which of the following statements provides the best definition of a simple event?

An event that is certain to happen.

An event that consists of more than one outcome.

An event that consists of exactly one outcome.

The collection of all possible outcomes of an experiment.
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Calculating theoretical probability for equally likely outcomes

Questions for: Calculating theoretical probability for equally likely outcomes

12

A single card is drawn from a standard deck of $52$ playing cards. What is the theoretical probability of drawing a face card (Jack, Queen, or King)?

$\frac{10}{13}$

$\frac{3}{13}$

$\frac{1}{13}$

$\frac{3}{52}$
13

A single fair six-sided die is rolled. What is the probability of rolling a prime number?

$\frac{1}{6}$

$\frac{1}{3}$

$\frac{1}{2}$

$\frac{2}{3}$
14

A bag contains $5$ red marbles, $8$ blue marbles, and $7$ green marbles. If one marble is drawn at random, what is the probability that it is not blue?

$\frac{7}{20}$

$\frac{1}{4}$

$\frac{2}{5}$

$\frac{3}{5}$
15

Two fair coins are tossed simultaneously. What is the probability of getting at least one head?

$\frac{2}{3}$

$\frac{1}{4}$

$\frac{1}{2}$

$\frac{3}{4}$
16

A spinner is divided into $8$ equal sectors, numbered $1$ through $8$. What is the probability of the spinner landing on a number that is a factor of $8$?

$\frac{1}{2}$

$\frac{1}{4}$

$\frac{3}{8}$

$\frac{5}{8}$
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Using the complement rule to find probabilities

Questions for: Using the complement rule to find probabilities

17

A fair six-sided die is rolled once. What is the probability of not rolling a number less than 3?

$1/3$

$2/3$

$1/6$

$1/2$
18

From a standard 52-card deck, one card is drawn at random. What is the probability that the card is not a face card (Jack, Queen, or King)?

$1/13$

$1/4$

$10/13$

$3/13$
19

In a school, 60% of the students have brown hair. If a student is chosen at random, what is the probability that the student does not have brown hair?

$0.30$

$0.50$

$0.60$

$0.40$
20

A fair coin is tossed three times. What is the probability of getting at least one head?

$3/8$

$1/8$

$7/8$

$1/2$
21

A bag contains 5 red marbles and 3 blue marbles. A marble is drawn at random, its color is noted, and then it is replaced. This process is repeated a second time. What is the probability of drawing at least one red marble?

$55/64$

$25/64$

$39/64$

$9/64$
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Calculating experimental probability from data

Questions for: Calculating experimental probability from data

22

A coin is tossed 150 times. It lands on heads 84 times. What is the experimental probability of the coin landing on tails?

$\frac{66}{84}$

$\frac{11}{25}$

$\frac{1}{2}$

$\frac{14}{25}$
23

A quality control inspector checks a batch of 500 light bulbs and finds 15 to be defective. What is the experimental probability that a randomly selected light bulb from this batch is not defective?

$\frac{15}{485}$

$\frac{3}{100}$

$\frac{97}{100}$

$\frac{1}{500}$
24

A survey asked 120 people about their favorite pizza topping. The results were: 45 chose Pepperoni, 30 chose Mushrooms, 25 chose Olives, and 20 chose Pineapple. What is the experimental probability that a randomly chosen person from this group prefers Pepperoni?

$\frac{1}{4}$

$\frac{3}{8}$

$\frac{1}{3}$

$\frac{3}{5}$
25

A spinner is spun 80 times. It landed on Red 24 times, Blue 30 times, Green 16 times, and Yellow 10 times. What is the experimental probability of the spinner not landing on Blue?

$\frac{1}{4}$

$\frac{3}{5}$

$\frac{3}{8}$

$\frac{5}{8}$
26

A student records the types of vehicles passing their school for an hour. They observe 60 Cars, 25 Trucks, 5 Buses, and 10 Motorcycles. Based on this data, what is the experimental probability that the next vehicle is a truck or a bus?

$\frac{1}{20}$

$\frac{1}{3}$

$\frac{3}{10}$

$\frac{1}{4}$
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Distinguishing between theoretical, experimental, and subjective probability

Questions for: Distinguishing between theoretical, experimental, and subjective probability

27

What type of probability is represented by the statement, "The probability of rolling a 4 on a fair six-sided die is $\frac{1}{6}$"?

Experimental probability

Theoretical probability

Subjective probability

Complementary probability
28

A quality control inspector tests 200 light bulbs from a production line and finds that 4 are defective. The inspector states that the probability of a randomly chosen bulb being defective is $\frac{4}{200} = \frac{1}{50}$. What kind of probability is this?

Experimental probability

Conditional probability

Theoretical probability

Subjective probability
29

A sports analyst says, "I have a gut feeling that the home team has about an 80% chance of winning tonight's game because their morale seems high." This statement is an example of which type of probability?

Experimental probability

Theoretical probability

Joint probability

Subjective probability
30

A coin is flipped 50 times, and it lands on tails 28 times. Which of the following statements represents a theoretical probability for this experiment?

The probability of the next flip being tails is higher than heads.

The probability of getting tails is $\frac{28}{50}$.

The probability of getting tails is $\frac{1}{2}$.

The probability of getting heads is $\frac{22}{50}$.
31

Which of the following scenarios is the BEST example of experimental probability?

The probability of a baby being a boy is approximately 0.51.

The probability of drawing a king from a standard 52-card deck is $\frac{4}{52}$.

To find the probability of a drawing pin landing point up, you toss it 100 times and record the results.

A doctor believes a new treatment has a 90% chance of success based on her experience.
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Using simulation to model and estimate probabilities

Questions for: Using simulation to model and estimate probabilities

32

What is the primary purpose of using a simulation in probability?

To estimate the probability of an event when it is difficult or impractical to calculate it theoretically or conduct the actual experiment.

To prove that a certain event is impossible.

To eliminate all randomness from an experiment.

To determine the exact theoretical probability of a complex event.
33

A professional archer hits the bullseye 80% of the time. Which of the following is the best simulation model for one of her shots?

Tossing a fair coin, where heads represents a bullseye.

Rolling a standard six-sided die, where rolling a number greater than 1 represents a bullseye.

Using a random number generator that produces integers from 1 to 10, where the numbers 1 through 8 represent a bullseye.

Drawing a card from a standard 52-card deck, where drawing a face card (Jack, Queen, King) represents a bullseye.
34

A weather forecast states there is a 20% chance of rain each day for the next three days. You want to simulate whether it rains on a given day. Which tool is most appropriate for this simulation?

A standard deck of 52 cards, where drawing a red card means 'Rain'.

A spinner divided into 5 equal sections, with one section labeled 'Rain'.

A standard six-sided die, where rolling a 1 or 2 means 'Rain'.

A fair coin, where heads means 'Rain'.
35

In a simulation to estimate the probability of winning a game, 200 trials were conducted. The 'Win' outcome occurred 58 times. Based on this simulation, what is the estimated probability of winning the game?

$58$

$0.58$

$0.71$

$0.29$
36

A simulation is run to estimate the probability that a family with three children will have exactly two girls. It is assumed $P(\text{Girl}) = P(\text{Boy}) = 0.5$. A coin is tossed three times for each trial (Heads = Girl, Tails = Boy). After 50 trials, the experimental probability is found to be $0.40$. The theoretical probability is $0.375$. What is the most likely reason for this difference?

A simulation can never perfectly match the theoretical probability.

Random variation due to a relatively small number of trials.

The coin used must have been biased.

The theoretical probability was calculated incorrectly.
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Foundations of Probability: Events, Rules, and Counting

1. Sample Spaces and Events

  • Sample Space (S): The set of all possible outcomes of an experiment. For example, when rolling a standard six-sided die, the sample space is $S = \{1, 2, 3, 4, 5, 6\}$.
  • Event (E): A subset of the sample space. It is a collection of one or more outcomes. For example, the event 'rolling an even number' is $E = \{2, 4, 6\}$.

2. Combining Events with Venn Diagrams

Venn diagrams are useful for visualizing relationships between events.

  • Union ($A \cup B$): The event that either A or B (or both) occurs. It is represented by the total area covered by both circles.

  • Intersection ($A \cap B$): The event that both A and B occur. It is represented by the overlapping area of the circles.

3. Mutually Exclusive Events

Two events, A and B, are mutually exclusive (or disjoint) if they cannot occur at the same time. This means their intersection is empty, i.e., $A \cap B = \emptyset$, and therefore $P(A \cap B) = 0$.

Example: When rolling a die, the event 'rolling a 2' and the event 'rolling a 5' are mutually exclusive. You can't roll both at once. In a Venn diagram, their circles would not overlap.

4. The Addition Rules of Probability

These rules help us find the probability of the union of events.

  • General Addition Rule: For any two events A and B, the probability that A or B occurs is:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

We subtract the intersection $P(A \cap B)$ because it is counted twice when we add $P(A)$ and $P(B)$.

  • Addition Rule for Mutually Exclusive Events:

If A and B are mutually exclusive, then $P(A \cap B) = 0$. The rule simplifies to:

$P(A \cup B) = P(A) + P(B)$

5. Multi-Stage Experiments

Many experiments happen in stages. We can use tools to find the total number of outcomes.

  • The Fundamental Counting Principle: If an event can occur in $m$ ways, and a second event can occur in $n$ ways, then the total number of ways both events can occur in sequence is $m \times n$.

Example: If you have 3 shirts and 2 pairs of pants, you have $3 \times 2 = 6$ possible outfits.

  • Tree Diagrams: A visual way to represent the sample space of a multi-stage experiment. Each branch represents a possible outcome for a stage.

Example: Tossing a coin twice. The first toss has 2 branches (H, T). From each of those, the second toss has 2 branches (H, T). The final outcomes are HH, HT, TH, TT.

6. Geometric Probability

For some problems, outcomes are represented by points in a geometric space (like a line, a rectangle, or a circle). Probability is found by comparing areas (or lengths).

$P(\text{Event}) = \frac{\text{Area of the favorable region}}{\text{Area of the total sample space}}$

Example: A dart is thrown at a square board with a circle inscribed in it. The probability of the dart landing in the circle is the ratio of the circle's area to the square's area.

37

For two events, A and B, from the same sample space, it is known that $P(A) = 0.6$, $P(B) = 0.5$, and the probability that both events occur is $P(A \cap B) = 0.3$. What is the probability that either event A or event B occurs, i.e., $P(A \cup B)$?

$0.3$

$0.8$

$1.1$

$0.2$
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Calculating geometric probabilities using area models

Questions for: Calculating geometric probabilities using area models

38

A square dartboard has a side length of 20 cm. A circular bullseye with a radius of 4 cm is in the center of the board. Assuming a dart that hits the board lands at a random point, what is the probability that it hits the bullseye?

$\frac{\pi}{100}$

$\frac{1}{25}$

$\frac{\pi}{25}$

$\frac{4\pi}{25}$
39

A point is selected at random inside a large rectangle with dimensions 15 units by 10 units. What is the probability that the point lies within a smaller, non-overlapping shaded square of side length 5 units located inside the rectangle?

$\frac{1}{6}$

$\frac{1}{4}$

$\frac{1}{3}$

$\frac{1}{5}$
40

A circular archery target has a total radius of 30 cm. The bullseye is a smaller circle in the center with a radius of 10 cm. If an arrow hits the target at a random point, what is the probability that it does not hit the bullseye?

$\frac{8}{9}$

$\frac{1}{9}$

$\frac{2}{3}$

$\frac{1}{3}$
41

A square is defined by the vertices A(0, 6), B(6, 6), C(6, 0), and D(0, 0) on a Cartesian plane. If a point is chosen randomly inside the square, what is the probability that it lies inside the triangle with vertices P(0, 0), Q(6, 0), and R(3, 3)?

$\frac{1}{3}$

$\frac{1}{2}$

$\frac{1}{4}$

$\frac{1}{6}$
42

A circular spinner is divided into four sectors with central angles: $45^\circ$ (Red), $75^\circ$ (Blue), $120^\circ$ (Green), and $120^\circ$ (Yellow). What is the probability that the spinner lands on a sector with a central angle of $120^\circ$ (i.e., either Green or Yellow)?

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{3}{4}$

$\frac{1}{2}$
43

A point is chosen at random from the interior of a circle. What is the probability that the point is closer to the center than to the circumference?

$\frac{1}{3}$

$\frac{1}{4}$

$\frac{\pi}{4}$

$\frac{1}{2}$
44

An airport runway is 2 kilometers long. A plane lands at a random point on the runway. What is the probability that the plane lands within the first 500 meters or the last 200 meters of the runway?

$\frac{1}{2}$

$\frac{3}{10}$

$\frac{1}{4}$

$\frac{7}{20}$
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Representing multi-stage experiments with tree diagrams

Questions for: Representing multi-stage experiments with tree diagrams

45

A fair coin is tossed twice. A tree diagram is drawn to represent all possible outcomes. How many branches will the second level of the tree diagram have in total?

4

8

1

2
46

A spinner with three equal sections labeled Red, Blue, and Green is spun once, and then a fair coin is tossed. What is the total number of possible outcomes for this two-stage experiment?

6

3

5

9
47

A bag contains 3 red marbles and 2 blue marbles. Two marbles are drawn one after another without replacement. What is the probability of drawing two red marbles?

$\frac{1}{10}$

$\frac{6}{25}$

$\frac{3}{10}$

$\frac{9}{25}$
48

A fair six-sided die is rolled, and then a coin is tossed. A tree diagram is used to list the sample space. Which set of outcomes represents the event of rolling an even number and then getting tails?

$\{(E, T)\}$

$\{(1, T), (3, T), (5, T)\}$

$\{(2, T), (4, T), (6, T)\}$

$\{(2, H), (4, H), (6, H)\}$
49

A box contains 2 green balls and 4 yellow balls. A ball is drawn, its color is noted, and then it is replaced. A second ball is then drawn. What is the probability that both balls drawn are green?

$\frac{4}{9}$

$\frac{1}{9}$

$\frac{1}{15}$

$\frac{1}{3}$
50

A student randomly guesses the answers to two true/false questions. A path on the tree diagram for this experiment goes from the start to 'Correct' on the first question, and then to 'Incorrect' on the second question. What is the probability of this specific path occurring?

$\frac{1}{3}$

$\frac{1}{2}$

$1$

$\frac{1}{4}$
51

A fair coin is tossed three times. Using a tree diagram, what is the probability of getting exactly two heads?

$\frac{1}{4}$

$\frac{1}{8}$

$\frac{1}{2}$

$\frac{3}{8}$
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Applying the fundamental counting principle

Questions for: Applying the fundamental counting principle

52

A restaurant offers a fixed-price menu with a choice of 4 different appetizers, 6 different main courses, and 3 different desserts. How many different three-course meals (one appetizer, one main course, one dessert) can be ordered?

$4 \times 6 + 3 = 27$

$4 + 6 + 3 = 13$

$6! \div 4! = 30$

$4 \times 6 \times 3 = 72$
53

A user is creating a passcode for their phone. The passcode must consist of one uppercase letter from the English alphabet followed by three digits (0-9). How many different passcodes can be created if repetition of digits is allowed?

$26 \times 10 \times 10 \times 10 = 26,000$

$26 \times 10 \times 9 \times 8 = 18,720$

$26 + 3 \times 10 = 56$

$26 + 10 + 10 + 10 = 56$
54

In how many different ways can 5 distinct books be arranged on a single shelf?

$5! = 120$

$5^2 = 25$

$5+4+3+2+1=15$

5
55

There are 3 roads from Town A to Town B and 5 roads from Town B to Town C. How many different routes are there to travel from Town A to Town C, passing through Town B?

$3 + 5 = 8$

$3 \times 5 = 15$

$5 - 3 = 2$

$3^5 = 243$
56

A student has 7 different shirts, 4 different pairs of pants, and 2 different pairs of shoes. Assuming any combination can be worn, how many different outfits consisting of one shirt, one pair of pants, and one pair of shoes can the student create?

$(7+4) \times 2 = 22$

$7 \times 4 \times 2 = 56$

$7 \times 4 + 2 = 30$

$7 + 4 + 2 = 13$
57

A standard license plate in a certain region consists of 3 letters followed by 3 digits. If any of the 26 letters and 10 digits can be used and repetition of both letters and digits is allowed, how many different license plates are possible?

$P(26,3) \times P(10,3) = 11,232,000$

$3 \times 26 + 3 \times 10 = 108$

$C(26,3) \times C(10,3) = 312,000$

$26^3 \times 10^3 = 17,576,000$
58

An experiment consists of flipping a fair coin once and then rolling a standard six-sided die twice. How many possible outcomes are there for this experiment?

$2 + 6 + 6 = 14$

$2 \times 6 = 12$

$2 \times 6 \times 6 = 72$

$2 + 6^2 = 38$
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Using Venn diagrams to represent the intersection and union of events

Questions for: Using Venn diagrams to represent the intersection and union of events

59

A Venn diagram represents two events, $A$ and $B$. Given that $P(A) = 0.5$, $P(B) = 0.4$, and the probability of their intersection is $P(A \cap B) = 0.2$, what is the probability of their union, $P(A \cup B)$?

$0.3$

$1.1$

$0.9$

$0.7$
60

In a Venn diagram, the region representing only event $C$ has a probability of $0.3$. The region representing only event $D$ has a probability of $0.4$. The region representing neither $C$ nor $D$ has a probability of $0.1$. What is the probability of the intersection of $C$ and $D$, denoted as $P(C \cap D)$?

$0.7$

$0.8$

$0.2$

$0.1$
61

A Venn diagram shows two overlapping circles representing events $A$ and $B$ within a universal set $S$. Which of the following descriptions corresponds to the event $A \cup B$?

The area outside both circles $A$ and $B$.

The area inside circle $A$ but outside circle $B$.

The entire area covered by both circle $A$ and circle $B$, including their overlap.

The area where the circles for $A$ and $B$ overlap.
62

In a Venn diagram with two events, $F$ and $G$, the shaded region where the two circles overlap represents which of the following events?

$G \setminus F$ (G only)

$F'$

$F \cup G$

$F \cap G$
63

In a class of 30 students, 18 play football (F) and 15 play basketball (B). If 7 students play both sports, how many students play at least one of the two sports? This corresponds to finding $n(F \cup B)$.

$26$

$19$

$4$

$33$
64

A Venn diagram for a sample space shows that the probability of the union of two events, $A$ and $B$, is $P(A \cup B) = 0.8$. What is the probability that neither event $A$ nor event $B$ occurs?

$0.2$

$1.8$

It cannot be determined from the given information.

$1.25$
65

Given two events $X$ and $Y$ in a sample space. A Venn diagram provides the following probabilities: $P(X \cup Y) = 0.9$, the probability of only event $Y$ occurring is $0.4$, and $P(X \cap Y) = 0.3$. What is the probability of event $X$, i.e., $P(X)$?

$0.3$

$0.5$

$0.2$

$0.6$
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Identifying mutually exclusive events

Questions for: Identifying mutually exclusive events

66

A single, fair six-sided die is rolled. Which of the following pairs of events is mutually exclusive?

Rolling a multiple of 3 and rolling an even number.

Rolling a number less than 3 and rolling a number greater than 4.

Rolling an even number and rolling a number greater than 4.

Rolling a prime number and rolling an odd number.
67

One card is drawn from a standard 52-card deck. Which of the following pairs of events is not mutually exclusive?

Drawing a Jack and drawing a Queen.

Drawing a Heart and drawing a King.

Drawing a red card and drawing a Spade.

Drawing a face card (Jack, Queen, or King) and drawing a 7.
68

In a school, let Event A be 'a randomly selected student is on the soccer team' and Event B be 'a randomly selected student is on the basketball team'. Under what condition are events A and B mutually exclusive?

Some students are on both teams.

Every student on the soccer team is also on the basketball team.

All students play at least one sport.

No student is on both the soccer and basketball teams.
69

A spinner is divided into 8 equal sectors, numbered 1 through 8. When the spinner is spun once, which pair of events is mutually exclusive?

The result is a number less than 5; the result is a number greater than 6.

The result is an even number; the result is a multiple of 4.

The result is a number greater than 3; the result is a multiple of 2.

The result is a prime number; the result is an odd number.
70

Two events, A and B, are defined as mutually exclusive. Which of the following mathematical statements must be true?

$P(A \cap B) = 0$

$P(A) + P(B) = 1$

$P(A) = P(B)$

$P(A \cup B) = 0$
71

A bag contains 5 red marbles, 4 blue marbles, and 3 green marbles. One marble is drawn at random. Identify the pair of mutually exclusive events.

Drawing a red marble; drawing a colored marble.

Drawing a red marble; drawing a marble that is not blue.

Drawing a green marble; drawing a marble that is not red.

Drawing a blue marble; drawing a green marble.
72

Consider two events related to tomorrow's weather: Event R is 'it will rain', and Event S is 'it will be sunny'. Are these events mutually exclusive in a real-world context?

No, because weather events are never mutually exclusive.

Yes, because the probability of rain and the probability of sun must add up to 1.

No, because phenomena like a 'sun shower' (rain while the sun is visible) can occur.

Yes, because it cannot be both rainy and sunny at the exact same time.
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Applying the General Addition Rule for probability

Questions for: Applying the General Addition Rule for probability

73

For two events, A and B, we are given that $P(A) = 0.6$, $P(B) = 0.5$, and $P(A \cap B) = 0.3$. What is the value of $P(A \cup B)$?

$0.8$

$0.4$

$1.1$

$0.2$
74

If $P(X) = 0.7$, $P(Y) = 0.4$, and $P(X \cup Y) = 0.8$, what is the probability of the intersection of events X and Y, $P(X \cap Y)$?

$0.1$

$0.5$

$0.3$

$1.1$
75

A single card is drawn from a standard 52-card deck. What is the probability that the card is a King or a Heart?

$\frac{17}{52}$

$\frac{3}{13}$

$\frac{1}{52}$

$\frac{4}{13}$
76

Two standard six-sided dice are rolled. What is the probability that the sum of the numbers is 8 or that at least one of the dice shows a 4?

$\frac{15}{36}$

$\frac{16}{36}$

$\frac{5}{36}$

$\frac{11}{36}$
77

In a class of 30 students, 18 take Physics, 15 take Chemistry, and 10 take both subjects. What is the probability that a randomly selected student takes Physics or Chemistry?

$\frac{33}{30}$

$\frac{13}{30}$

$\frac{7}{30}$

$\frac{23}{30}$
78

Given two events A and B where $P(A') = 0.4$, $P(B') = 0.3$, and $P(A \cap B) = 0.5$. Find the probability of $P(A \cup B)$.

$0.2$

$1.3$

$0.8$

$0.6$
79

In a survey, it was found that 40% of people have brown hair, 25% have brown eyes, and 15% have both brown hair and brown eyes. What is the probability that a randomly selected person has brown hair or brown eyes?

$0.80$

$0.50$

$0.10$

$0.65$
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Applying the Addition Rule for mutually exclusive events

Questions for: Applying the Addition Rule for mutually exclusive events

80

A single card is drawn from a standard 52-card deck. What is the probability of drawing a King or an Ace?

$2/13$

$1/13$

$4/13$

$1/169$
81

A fair six-sided die is rolled once. What is the probability of rolling a number less than 3 or a number greater than 4?

$2/3$

$5/6$

$1/9$

$1/3$
82

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If one marble is drawn at random, what is the probability that it is blue or green?

$3/10$

$1/5$

$1/4$

$1/2$
83

In a group of 100 students, 25 are in the debate club and 30 are in the chess club. No student is in both clubs. What is the probability that a randomly selected student is in the debate club or the chess club?

$3/10$

$11/20$

$1/4$

$3/40$
84

A spinner is divided into 8 equal sectors, numbered 1 to 8. What is the probability that the spinner lands on an even number or on the number 7?

$1/8$

$1/16$

$1/2$

$5/8$
85

A letter is chosen at random from the word PROBABILITY. What is the probability of choosing the letter 'B' or a vowel?

$4/11$

$8/121$

$6/11$

$2/11$
86

Events A and B are mutually exclusive. If $P(A) = 0.3$ and $P(B) = 0.4$, what is $P(A \text{ or } B)$?

$1.0$

$0.1$

$0.7$

$0.12$

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