Respuesta :
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
[tex]P(X<6)=P(X\leq 5)=0.26[/tex]
[tex]P(6\leq X\leq 20)=0.53[/tex]
To find : The probability of each of the following outcomes.
a) A student makes more than 20 mistakes
i.e. [tex]P(X>20)[/tex]
[tex]P(X>20)=1-P(X\leq 20)[/tex]
[tex]P(X>20)=1-(P(X\leq 5)+P(6\leq X\leq 20))[/tex]
[tex]P(X>20)=1-(0.26+0.53)[/tex]
[tex]P(X>20)=1-(0.79)[/tex]
[tex]P(X>20)=0.21[/tex]
b. A student makes 6 or more mistakes
i.e. [tex]P(X\geq 6)=1-P(X<6)[/tex]
[tex]P(X\geq 6)=1-0.26[/tex]
[tex]P(X\geq 6)=0.74[/tex]
c. A student makes at most 20 mistakes
i.e. [tex]P(X\leq 20)=1-P(X>20)[/tex]
Using 'a' part [tex]P(X>20)=0.21[/tex]
[tex]P(X\leq 20)=1-0.21[/tex]
[tex]P(X\leq 20)=0.79[/tex]
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.
Answer:
(a) P(X > 20) = 0.18
(b) P([tex]X\geq 6) = 0.71[/tex]
(c) P([tex]X\leq 20) = 0.82[/tex]
(d) Events (a) and (c)
Step-by-step 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([tex]X \leq 5[/tex] = 0.29
P([tex]6\leq X\leq 20[/tex]) = 0.53
Now,
(a) When a student makes more than 20 mistakes:
P(X > 20) = 1 - P([tex]X\leq 20[/tex])
P(X > 20) = 1 - {P([tex]X\leq 5) + P(6\leq X\leq 20)[/tex]}
P(X > 20) = 1 - {0.29 + 0.53) = 0.18
(b) When the student makes 6 mistakes or more:
P([tex]X\geq 6 = 1 - P(X\leq 5) = 1 - 0.29 = 0.71[/tex]
(c) When the student makes at most 20 mistakes:
P([tex]X\leq 20) = 1 - P(X > 20) = 1 - 0.18 = 0.82[/tex]
(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.
