Question

Ed and Taylor work together on a group quiz that has 20 multiple-choice questions, each with four choices for the possible answer. Unfortunately, neither had time to study, so they decide to randomly guess at all answers. Ed guesses answers for the first 10 questions, and Taylor guesses for the other 10 questions.

a. For any single question on the quiz, what is p = chance of a correct guess?
b. Let X = number of correct guesses that Isabelle makes. What are the values of n and p for the binomial distribution that describes X?
c. Let Y = number of correct guesses that Taylor makes. What are the values of n and p for the binomial distribution that describes Y?
d. Consider X + Y = total correct guesses that Isabelle and Taylor make. What are the values of n and p for the binomial distribution that describes X + Y?

Answer 1

a. p = 0.25

b. n = 10, p = 0.25.

c. n = 10, p = 0.25.

d. n = 20, p = 0.25.

Explanation:

Binomial probability distribution

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

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

a. For any single question on the quiz, what is p = chance of a correct guess?

4 options, 1 of which is correct. So

`p = (1)/(4) = 0.25`

So p = 0.25.

b. Let X = number of correct guesses that Isabelle makes. What are the values of n and p for the binomial distribution that describes X?

Should be Ed here.

Ed guesses answers for the first 10 questions

This means that
`n = 10`.

p is the same, so n = 10, p = 0.25.

c. Let Y = number of correct guesses that Taylor makes. What are the values of n and p for the binomial distribution that describes Y?

10 question, so
`n = 10`, p is the same.

d. Consider X + Y = total correct guesses that Isabelle and Taylor make. What are the values of n and p for the binomial distribution that describes X + Y?

20 questions, so
`n = 20`

10 for Ed, with 0.25 probability, 10 for Taylor, with 0.25 probability. So

`p = (10)/(20)*0.25 + (10)/(20)*0.25 = 0.125 + 0.125 = 0.25`

So n = 20, p = 0.25.