Question

Casey ran out of time while taking a multiple-choice test and plans to guess on the last \[10\] questions. Each question has \[5\] possible choices, one of which is correct. Let \[X=\] the number of answers Casey correctly guesses in the last \[10\] questions. Which of the following would find \[P(X=2)\]? Choose 1 answer: Choose 1 answer: (Choice A) \[\left( \dfrac{1}5 \right)^2\left( \dfrac{4}5 \right)^8\] A \[\left( \dfrac{1}5 \right)^2\left( \dfrac{4}5 \right)^8\] (Choice B) \[\displaystyle{10 \choose 5} \left( \dfrac{1}5 \right)^2\left( \dfrac{4}5 \right)^8\] B \[\displaystyle{10 \choose 5} \left( \dfrac{1}5 \right)^2\left( \dfrac{4}5 \right)^8\] (Choice C) \[\displaystyle{10 \choose 5} \left( \dfrac{1}5 \right)^8\left( \dfrac{4}5 \right)^2\] C \[\displaystyle{10 \choose 5} \left( \dfrac{1}5 \right)^8\left( \dfrac{4}5 \right)^2\] (Choice D) \[\displaystyle{10 \choose 2} \left( \dfrac{1}5 \right)^2\left( \dfrac{4}5 \right)^8\] D \[\displaystyle{10 \choose 2} \left( \dfrac{1}5 \right)^2\left( \dfrac{4}5 \right)^8\] (Choice E) \[\displaystyle{10 \choose 2} \left( \dfrac{1}5 \right)^8\left( \dfrac{4}5 \right)^2\] E \[\displaystyle{10 \choose 2} \left( \dfrac{1}5 \right)^8\left( \dfrac{4}5 \right)^2\]

Answer 1

To find the probability that Casey correctly guesses 2 out of 10 multiple-choice questions, use the binomial probability formula. The correct choice is D which uses 10C2 * (1/5)^2 * (4/5)^8.

Step-by-step explanation:

The probability that Casey correctly guesses exactly 2 out of the last 10 questions on a multiple-choice test, given that each question has 5 possible answers, can be found using the binomial probability formula.

This formula is P(X = k) = nCk * (p)^k * (q)^(n-k) where nCk represents the number of combinations of n items taken k at a time, p is the probability of success on a single trial, and q is the probability of failure. Since Casey is guessing and there is one correct answer out of five choices, p = 1/5 and q = 4/5.

The formula is P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where P(X=k) is the probability of getting k successes, C(n, k) is the number of ways to choose k successes out of n trials, p is the probability of success on a single trial, and (1-p) is the probability of failure on a single trial.

In this case, n=10 (number of trials), k=2 (number of successes), p=1/5 (probability of guessing the correct answer), and (1-p)=4/5 (probability of guessing the incorrect answer). Substituting these values into the formula, we get P(X=2) = C(10, 2) * (1/5)^2 * (4/5)^8

For 2 correct guesses out of 10 questions, the calculation is 10C2 * (1/5)^2 * (4/5)^8, which corresponds to Choice D.