Question

Assume that random guesses are made for 9 multiple choice questions on an SAT​ test, so that there are n=9 ​trials, each with probability of success​ (correct) given by p= .25. Find the indicated probability for the number of correct answers. Find the probability that the number x of correct answers is fewer than 4

Answer 1

To find the probability that the number x of correct answers is fewer than 4, you can use the binomial probability formula. In this case, you want to find P(x < 4) for n = 9 trials and p = 0.25 (the probability of success).

P(x < 4) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3)

Using the binomial probability formula:

P(x = k) = (n choose k) * p^k * (1 - p)^(n - k)

Where (n choose k) is the binomial coefficient, given by:

(n choose k) = n! / (k! * (n - k)!)

Let's calculate each term:

P(x = 0):

(9 choose 0) * (0.25)^0 * (0.75)^(9 - 0) = 1 * 1 * 0.75^9

P(x = 1):

(9 choose 1) * (0.25)^1 * (0.75)^(9 - 1) = 9 * 0.25 * 0.75^8

P(x = 2):

(9 choose 2) * (0.25)^2 * (0.75)^(9 - 2) = 36 * 0.25^2 * 0.75^7

P(x = 3):

(9 choose 3) * (0.25)^3 * (0.75)^(9 - 3) = 84 * 0.25^3 * 0.75^6

Now, calculate each of these probabilities:

P(x < 4) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3)

P(x < 4) = (1 * 1 * 0.75^9) + (9 * 0.25 * 0.75^8) + (36 * 0.25^2 * 0.75^7) + (84 * 0.25^3 * 0.75^6)

Calculate these values, and you will find the probability that the number of correct answers is fewer than 4.

Explanation:

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