Question

A driving exam consists of 29 ​multiple-choice questions. Each of the 29 answers is either right or wrong. Suppose the probability that a student makes fewer than 6 mistakes on the exam is 0.26 and that the probability that a student makes from 6 to 20 ​(inclusive) mistakes is 0.53. Find the probability of each of the following outcomes. a. A student makes more than 20 mistakes b. A student makes 6 or more mistakes c. A student makes at most 20 mistakes d. Which two of these three events are​ complementary?

Answer 1

(a) P(X > 20) = 0.18

(b) P(
`X\geq 6) = 0.71`

(c) P(
`X\leq 20) = 0.82`

(d) Events (a) and (c)

Explanation:

As per the question:

Total no. of multiple choice questions = 29

Now,

Let the no. of mistakes that a student make be X.

Then

P(X < 6) = P(
`X \leq 5` = 0.29

P(
`6\leq X\leq 20`) = 0.53

Now,

(a) When a student makes more than 20 mistakes:

P(X > 20) = 1 - P(
`X\leq 20`)

P(X > 20) = 1 - {P(
`X\leq 5) + P(6\leq X\leq 20)`}

P(X > 20) = 1 - {0.29 + 0.53) = 0.18

(b) When the student makes 6 mistakes or more:

P(
`X\geq 6 = 1 - P(X\leq 5) = 1 - 0.29 = 0.71`

(c) When the student makes at most 20 mistakes:

P(
`X\leq 20) = 1 - P(X > 20) = 1 - 0.18 = 0.82`

(d) The two complementary events are (a) and (c), i.e., the event when a student more than 20 mistakes and when at most 20 mistakes are made by the student.

Answer 2

Answer and explanation:

Given : A driving exam consists of 29 ​multiple-choice questions. Each of the 29 answers is either right or wrong. Suppose the probability that a student makes fewer than 6 mistakes on the exam is 0.26 and that the probability that a student makes from 6 to 20 ​(inclusive) mistakes is 0.53.

Let X be the number of mistake

`P(X<6)=P(X\leq 5)=0.26`

`P(6\leq X\leq 20)=0.53`

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

a) A student makes more than 20 mistakes

i.e.
`P(X>20)`

`P(X>20)=1-P(X\leq 20)`

`P(X>20)=1-(P(X\leq 5)+P(6\leq X\leq 20))`

`P(X>20)=1-(0.26+0.53)`

`P(X>20)=1-(0.79)`

`P(X>20)=0.21`

b. A student makes 6 or more mistakes

i.e.
`P(X\geq 6)=1-P(X<6)`

`P(X\geq 6)=1-0.26`

`P(X\geq 6)=0.74`

c. A student makes at most 20 mistakes

i.e.
`P(X\leq 20)=1-P(X>20)`

Using 'a' part
`P(X>20)=0.21`

`P(X\leq 20)=1-0.21`

`P(X\leq 20)=0.79`

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.