Multiple Choice

1. For the binomial distribution shown by $\Pr(X = x) = {6 \choose x} (0.3)^x (0.7)^{6-x}$, what are the values for the number of trials ($n$) and the probability of success ($p$)?

   $n=7$ and $p=0.3$

   $n=6$ and $p=0.7$

   $n=6$ and $p=0.3$

   $n=x$ and $p=0.3$

2. To find the value of $\Pr(X=4)$, as requested in part (b), which expression correctly represents the substitution of $x=4$ into the given binomial formula?

   ${6 \choose 4} (0.3)^2 (0.7)^4$

   ${6 \choose 4} (0.3)^4 (0.7)^2$

   ${4 \choose 6} (0.3)^4 (0.7)^2$

   ${6 \choose 6} (0.3)^4 (0.7)^2$

3. The problem describes a binomial experiment involving rolling a fair die 60 times. If the event of 'success' is defined as observing a 6, which options correctly identify the number of trials ($n$) and the probability of success ($p$)?

   $n=6$ and $p=1/6$

   $n=60$ and $p=1/5$

   $n=10$ and $p=1/6$

   $n=60$ and $p=1/6$

4. If $X$ represents the random variable for the total number of 6s observed in the 60 rolls, which mathematical expression is equivalent to finding the probability of observing \"fewer than ten 6s\" (part b)?

   $P(X \ge 10)$

   $P(X \le 9)$

   $P(X = 10)$

   $P(X > 10)$

5. The rainfall records for the city of Melbourne indicate that the probability of rain falling on any one day in November is $0.35$. What is the probability that it will NOT rain on a specific, randomly chosen day in November?

   $0.65$

   $0.35$

   $0.75$

   $0.50$

6. Given the probability of rain on any day is $0.35$, and assuming the occurrence of rain on any day is independent, calculate the probability that it rains on both the first day and the second day of a given week.

   $0.4225$

   $0.105$

   $0.70$

   $0.1225$

7. A standard six-sided die is rolled seven times. Calculate the probability that the first roll results in a 2 and the remaining six rolls do not result in a 2, as described in part 'a'.

   $7 * (1/6) * (5/6)^6$

   $(1/6)^7$

   $(1/6) * (5/6)^6$

   $(1/6) * (1/6)^6$

8. The problem asks to find the probability that exactly one of the seven rolls results in a 2 (part 'b'). If $X$ is the number of 2s obtained, which of the following expressions correctly represents $P(X=1)$? (Note: $C(n, k)$ represents the number of combinations.)

   $C(7, 1) * (1/6)^1 * (5/6)^6$

   $(1/6) * (5/6)^6$

   $C(7, 1) * (1/6)^1 * (5/6)^7$

   $C(7, 6) * (1/6)^6 * (5/6)^1$

9. In the context of the binomial distribution described in the problem, where $n$ is the total number of independent trials and $p$ is the constant probability of a female child ('success'), what are the required values for $n$ and $p$?

   $n = 100$ and $p = 0.6$

   $n = 50$ and $p = 0.5$

   $n = 100$ and $p = 0.5$

   $n = 60$ and $p = 0.5$

10. If $X$ is the random variable representing the number of female babies born, the requirement that \"more than 60 of them will be female\" must be translated into which mathematical probability statement?

    $P(X > 60)$

    $P(X \le 60)$

    $P(X = 60)$

    $P(X < 60)$

11. A breakfast cereal manufacturer places a coupon in every tenth packet of cereal. If a family purchases a single packet, what is the probability ($p$) that this specific packet contains a coupon?

    $0.1$

    $0.5$

    $0.01$

    $0.9$

12. A family purchases five packets of cereal. The probability of finding a coupon in any single packet is $p=0.1$. If we calculate the probabilities for the number of coupons found ($X$), we find that $P(X=0) \approx 0.590$ and $P(X=1) \approx 0.328$. Based on this distribution, what is the most probable number of coupons the family will find in the five packets?

    1 coupon

    0 coupons

    5 coupons

    2 coupons

13. Given that there are four independent checkouts, and the probability of any single checkout being free is $0.25$, calculate the probability that the customer makes a purchase (meaning at least one checkout is free). Round your answer to four decimal places.

    $0.0039$

    $0.3125$

    $0.6836$

    $0.7500$

14. If the customer only decides to leave without a purchase when all four checkouts are busy, what is the probability of this specific outcome occurring? Express your answer as a fraction in simplest form.

    $1/256$

    $81/256$

    $175/256$

    $3/4$

15. An aircraft has four engines. The probability that any single engine will fail is $0.003$. If all engines operate independently, what is the probability (rounded to five decimal places) that 'no engine failure occurs' (scenario a)?

    $0.99700$

    $0.98805$

    $0.98800$

    $0.01200$

16. An aircraft has four independent engines, and the probability of any one engine failing is $0.003$. Determine the probability that 'all four engines fail' (scenario c).

    $8.1 \times 10^{-9}$

    $0.000081$

    $8.1 \times 10^{-11}$

    $0.012$

17. It has been discovered that $4\%$ of the batteries produced at the factory are defective. If a sample of 10 batteries is drawn randomly, what is the expected number of defective batteries in this sample?

    $4$ batteries

    $0.4$ batteries

    $10$ batteries

    $0.04$ batteries

18. The factory typically has a $4\%$ defect rate. If a particular sample of 10 batteries contained six defective batteries, what is the most reasonable statistical doubt one might have regarding the factory's operation during that hour?

    The method used to draw the random sample was flawed and improperly conducted.

    The defect rate likely decreased significantly below $4\%$ immediately after this sample was drawn.

    The sample size of 10 was too small to accurately reflect the true $4\%$ defect rate.

    The defect rate for that specific hour might have been temporarily much higher than the usual $4\%$.

19. An examination consists of multiple-choice questions, where each question has four possible answers. If a student randomly guesses the answer to a single question, what is the probability that the guess is correct?

    $1/10$

    $2/4$

    $3/4$

    $1/4$

20. The examination consists of 10 multiple-choice questions. A student passes if they get at least $50\%$ of the answers correct. What is the minimum number of correct answers required to pass the entire examination?

    4

    10

    6

    5