Question

A.The Murray family has produced 8 children.

What is the probability that there are an equal number of boys and girls?


B.Margie has the uncanny ability of being able to guess the correct answer to TRUE/FALSE questions with an 80% accuracy rate. Tomorrow’s test has 7 questions on it and she needs to answer 5 questions correctly in order to make the Honor Roll. If she guesses on every question, what is the probability that she’ll answer exactly 5 questions correctly?

Answer 1

A. 0.2734 = 27.34% probability that there are an equal number of boys and girls

B. 0.2753 = 27.53% probability that she’ll answer exactly 5 questions correctly

Explanation:

In each of these questions, the binomial probability distribution is used.

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.The Murray family has produced 8 children.

What is the probability that there are an equal number of boys and girls?

8 children means that
`n = 8`

Each children is equally as likely to be a boy or a girl, which means that
`p = 0.5`

This probability is P(X = 4). So

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

`P(X = 4) = C_(8,4).(0.5)^(4).(0.5)^(4) = 0.2734`

0.2734 = 27.34% probability that there are an equal number of boys and girls

B.Margie has the uncanny ability of being able to guess the correct answer to TRUE/FALSE questions with an 80% accuracy rate. Tomorrow’s test has 7 questions on it and she needs to answer 5 questions correctly in order to make the Honor Roll. If she guesses on every question, what is the probability that she’ll answer exactly 5 questions correctly?

80% accurracy rate means that
`p = 0.8`

7 questions means that
`n = 7`

The probability is P(X = 5). So

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

`P(X = 5) = C_(7,5).(0.8)^(5).(0.2)^(2) = 0.2753`

0.2753 = 27.53% probability that she’ll answer exactly 5 questions correctly