Bill has completed a 10-question multiple-choice test on which he answered 7 questions correctly. Each question had one correct answer to be chosen from five alternatives. Bill says that he answered the test by randomly guessing the answers without reading the questions or answers. Define the random variable x to be the number of correct answers on this test, and construct the probability distribution if the answers were obtained by random guessing.

Question

Bill has completed a 10-question multiple-choice test on which he answered 7 questions correctly. Each question had one correct answer to be chosen from five alternatives. Bill says that he answered the test by randomly guessing the answers without reading the questions or answers. Define the random variable x to be the number of correct answers on this test, and construct the probability distribution if the answers were obtained by random guessing.

What is the probability that Bill guessed 7 of the 10 answers correctly?

Answer 1

cutting through the chase ....

prob(your stated event)
= C(10,7)(1/5)^7 (4/5)^3
= 120(1/78125)(64/125) = appr .0008

Answer 2

To find the probability that Bill guessed 7 of the 10 answers correctly, we need to use the binomial distribution formula.

The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial (e.g., success or failure), with a fixed number of trials and each trial having the same probability of success.

In this case, the random variable x represents the number of correct answers given by Bill, with 7 being the desired outcome.

Let's break down the components of the binomial distribution formula:

1. n represents the number of trials. Here, the number of trials is 10, as there are 10 questions.

2. p represents the probability of success in a single trial. Since Bill guessed the answers randomly, there is a 1/5 chance of guessing each question correctly. Therefore, p = 1/5.

3. q represents the probability of failure in a single trial. In this case, q would be 1 - p, which is 4/5.

Now, let's calculate the probability using the binomial distribution formula:

P(x=k) = nCk * p^k * q^(n-k)

P(x=7) = 10C7 * (1/5)^7 * (4/5)^(10-7)

To calculate 10C7 (pronounced "10 choose 7"), you can use the formula for combinations:

10C7 = 10! / (7! * (10-7)!) = 10! / (7! * 3!)

Calculating these values, we get:

10C7 = 10*9*8 / (3*2*1) = 120

Now, substituting all the values into the formula:

P(x=7) = 120 * (1/5)^7 * (4/5)^3

Simplifying further:

P(x=7) = 120 * (1/78125) * (64/125)

Final calculation:

P(x=7) = 120 * 64 / (78125 * 125)

The resulting value is the probability that Bill guessed 7 of the 10 answers correctly.