A state end-of-grade exam in American History is a multiple-choice test that has 50 questions with 4 answer choices for each question. A student must get at least 25 correct to pass the test, and the questions are very difficult. Question 1. If a student guesses on every question, what is the probability the student will pass? (round your answer to 4 decimal places). Question 2. Suppose, after studying, a student raises her chances of getting each question correct to 0.70. What is the probability that she will pass? (round your answer to 4 decimal places).

For each question, there are only two possible outcomes. Either he gets it correct, or he gets it wrong. So we solve this problem using the binomial probability distribution.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.\pi^{x}.(1-\pi)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinatios of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And [tex]\pi[/tex] is the probability of X happening.

For this problem, we have that:

Question 1.

There are 50 questions, so [tex]n = 50[/tex].

The student is going to guess each question, so he has a [tex]\pi = \frac{1}{4} = 0.25[/tex] probability of getting it right.

He needs to get at least 25 question right.

So we need to find [tex]P(X \geq 25)[/tex].

Using a binomial probability calculator, with [tex]n = 50[/tex] and [tex]\pi = 0.25[/tex] we get that [tex]P(X \geq 25) = 0.0001[/tex].

This means that the student has a 0.01% probability of passing the test.

Question 2.

Now, we need to find [tex]P(X \geq 25)[/tex] with [tex]\pi = 0.70[/tex]. So [tex]P(X \geq 25) = 0.9991[/tex]

She has a 99.91% probability of passing in the test.

sidratulmuntaha7 sidratulmuntaha7 · Answer 2 · 2025-03-22T16:28:33-04:00

Answer:

Q1) The student has a 0.01% probability of passing the test.

Q2) She has a 99.91% probability of passing in the test.

Step-by-step explanation:

For each question, there are only two possible outcomes. Either he gets it correct, or he gets it wrong. So we solve this problem using the binomial probability distribution.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

In which is the number of different combinatios of x objects from a set of n elements, given by the following formula.

And is the probability of X happening.

For this problem, we have that:

Question 1.

There are 50 questions, so .

The student is going to guess each question, so he has a probability of getting it right.

He needs to get at least 25 question right.

So we need to find .

Using a binomial probability calculator, with and we get that .

This means that the student has a 0.01% probability of passing the test.

Question 2.

Now, we need to find with . So

She has a 99.91% probability of passing in the test.

Step-by-step explanation: