Susan Nall

Answered

2021-12-17

Beth is taking an eleven-question multiple-choice test for which each question has three answer choices, only one of which is correct. Beth decides on answers by rolling a fair die and marking the first answer choice if the die shows 1 or 2, the second if the die shows 3 or 4, and the third if the die shows 5 or 6. Find the probability of the stated event.

exactly

four

correct answers

exactly

four

correct answers

Answer & Explanation

Jenny Bolton

Expert

2021-12-18Added 32 answers

Step 1

Introduction:

In a fair die, there are 6 possible outcomes, 1, 2, ..., 6, all of which are equally likely, that is, with probability

Step 2

Calculation:

On rolling the fair die once, the probability of getting 1 or 2 is

Similarly, the probability of getting 3 or 4 is 1/3, and the probability of getting 5 or 6 is 1/3.

Thus, in case of each of the 11 questions, the probability of selecting each option is 1/3.

As the outcomes on the roll of a fair die are independent from one roll to another, the choices made in the different questions are also independent of one another.

Considering each attempted question as a trial, there are

Considering it to be a success if the correct option is chosen, the probability of success in each trial is

Here, X can be considered as the number of successes, that is, number of questions correctly answered. Thus, X has a binomial distribution with parameters,

Since p denotes the probability of success, the probability of failure is,

If

The probability mass function of X here is:

The probability of exactly 4 correct answers, that is, exactly 4 successes is calculated below:

Thus, the probability of exactly 4 correct answers is 0.238446.

Debbie Moore

Expert

2021-12-19Added 43 answers

Step 1

Binomial Problem with

Find the probability of

a) exactly four correct answers

b) fewer than three correct answers

nick1337

Expert

2021-12-28Added 573 answers

Step 1

Hence, the probability of getting the correct answer is

Note that the probability of x successes out of n trials is

where

Thus, the probability is

P(6)=0.017070569

Most Popular Questions