Question

A driving exam consists of 26 ​multiple-choice questions. Each of the 26 answers is either right or wrong. Suppose the probability that a student makes fewer than 7 mistakes on the exam is 0.29 and that the probability that a student makes from 7 to 18 ​(inclusive) mistakes is 0.58. Find the probability of each of the following outcomes. a. A student makes more than 18 mistakes b. A student makes 7 or more mistakes c. A student makes at most 18 mistakes d. Which two of these three events are​ complementary?

Answer 1

Answer and explanation:

Given : A driving exam consists of 26 ​multiple-choice questions. Each of the 26 answers is either right or wrong. Suppose the probability that a student makes fewer than 7 mistakes on the exam is 0.29 and that the probability that a student makes from 7 to 18 ​(inclusive) mistakes is 0.58.

Let X be the number of mistake

`P(X<7)=P(X\leq 6)=0.29`

`P(7\leq X\leq 18)=0.58`

To find : The probability of each of the following outcomes.

a) A student makes more than 18 mistakes

i.e.
`P(X>18)`

`P(X>18)=1-P(X\leq 18)`

`P(X>18)=1-(P(X\leq 6)+P(7\leq X\leq 18))`

`P(X>18)=1-(0.29+0.58)`

`P(X>18)=1-(0.87)`

`P(X>18)=0.13`

b. A student makes 7 or more mistakes

i.e.
`P(X\geq 7)=1-P(X<7)`

`P(X\geq 7)=1-0.29`

`P(X\geq 7)=0.71`

c. A student makes at most 18 mistakes

i.e.
`P(X\leq 18)=1-P(X>18)`

Using 'a' part
`P(X>18)=0.13`

`P(X\leq 18)=1-0.13`

`P(X\leq 18)=0.87`

d. Which two of these three events are​ complementary?

The complement of an event happening is the exact opposite: the probability of it not happening.

According to definition,

Option a and c are complementary events.