3.1 - 3.3 Problems:
The sides of a six-sided spinner are numbered from 1 to 6. The table shows the results
for 100 spins.
Number on Spinner
1
2
3
4
5
6
Frequency
27
18
17
15
16
7
a. what is the relative frequency of getting a one?
b. Do you think the spinner is fair? Give a reason for your answer.
c. The spinner is spun 3000 times. Estimate the number of times the result will be a 4.
In a group of 35 children, 10 have blonde hair, 14 have brown eyes, and 4 have both
blonde hair and brown eyes. Draw a Venn diagram to represent this situation.
A child is selected at random from the class. What is the probability that the child has
blond hair or brown eyes?
Draw a sample space diagram for the random experiment ‘two tetrahedral dice, one blue
and the other red, are each numbered 1 to 4. They are rolled and the result noted’.
FInd the probability that:
a. the number on the red dice is greater than the number on the blue dice
b. the difference between the numbers on the dice is one
c. the red dice shows an odd number and the blue dice shows an even number
d. the sum of the numbers on the dice is prime
3.4 Problem:
There are 27 students in a class. 15 take Art and 20 take Theater. Four do neither subject.
How many students do both subjects? One person is chosen at random. Find the
probability that
a. he or she takes Theater but not Art
b. he or she takes at least one of the two subjects
c. he or she takes Theater, given that he or she takes Art
Solution:
Let n(A ∩ D) = x
15 − x + x + 20 − x + 4 = 27
39 − x = 27
x =12
a. P(Drama not Art) =
𝟖
𝟐𝟕
b. P(Takes at least one of the two subjects) = 1 – P(takes none) = 1 − 4/27 = 23/27
c. P(Takes both subjects, given that he takes Art) = 12/27/15/27 = 12/15 = 4/5
8.5 Problem:
The table below shows the cumulative frequency distribution for the times taken by 100
students to eat lunch
Time (min)
Number of Students
2 and under
0
4 and under
6
6 and under
18
8 and under
24
10 and under
40
12 and under
60
14 and under
78
16 and under
92
18 and under
100
Using a scale of 1cm and 10 students on the vertical axis and 1cm for 2 minutes on hte
horizontal axis, plot and draw a cumulative frequency diagram
Use the graph to estimate i. the median ii. the interquartile range
Solution:
Following directions given, make a cumulative frequency graph
i. 11 mins
ii. (13.6–8.2) mins = 5.4 mins.
8.6 Problems:
The number of children in the families in a class of 29 children is shown below. Find the
mean and the standard deviation.
Children
1
2
3
4
5
6
7
f
5
12
8
3
0
0
1
Solution:
Children
f
(fx)
1
5
5
2
12
24
3
8
24
4
3
12
5
0
0
6
0
0
7
1
7
=29
72/29= 2.5
=72
The mean is 2.5
(1-2.5)^2 = 2.25
(2-2.5)^2 = 0.25
(2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 2.25) / 7 = 6.25
(3-2.5)^2 = 0.25
(4-2.5)^2 = 2.25
(5-2.5)^2 = 6.25
(6-2.5)^2 = 12.25
(7-2.5)^2 = 20.25
√𝟔. 𝟐𝟓= Standard deviation
15.1 Problem/Solution:
The random variable X has the probability distribution
X
1
2
3
4
5
P(X = x)
7c
5c
4c
3c
c
a) Find the value of c
7c + 5c + 4c + 3c + c = 1
20c = 1
c = 1/20
b) Find P(X ≥ 4)
(greater than or equal to 4)
P(X ≥ 4) = P(X = 4) + P(X = 5)
= 3/2 + 1/20 =4/20
= 1/5
15.2 Problems:
1. X is binomially distributed with 4 trials and a probability of success equal to ½ on each
trial. Without a calculator determine the probability of:
a. P(X=1)
b. P(X<1)
c. P(X<1)
d. P(X>1)
15.3 Problems:
Given that Z ~ N(0,1) find
a. P(-1<Z<1)
b. P(-2<Z<2)
c. P(-3<Z<3)