Find the probability that the number x of correct answers is fewer than 4.

Question

Assume that random guesses are made for seven multiple choice questions on an SAT​ test, so that there are

n=7 ​trials, each with probability of success​ (correct) given by p=0.350 Find the indicated probability for the number of correct answers.

Answer 1

This is an example of Bernouli trials so we can use the Binomial Distribtution to answer this.

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

Remember that, for binomial distributions with probability of success p, number of trials n, the equation is:

P(X = r) = _nC_r * p^r * (1 - p)^n-r

P(X = 0) = (1 - .35)⁷ = .0490

P(X = 1) = 7 * (.35)¹ * (1 - .35)⁶ = .1848

P(X = 2) = ₇C₂ * (.35)² * (1 - .35)⁵ = .2985

P(X = 3) = ₇C₃ * (.35)³ * (1 - .35)⁴ = .2679

So, P(X < 4) = .8002.

This is reasonable, if there is only a 35% chance of getting a question right, it is likely that you will get fewer than half the questions right.