Question

Isabel Briggs Myers was a pioneer in the study of personality types. The personality types are broadly defined according to four main preferences. Do married couples choose similar or different personality types in their mates? The following data give an indication.

Similarities and Differences in a Random Sample of 375 Married Couples
Number of Similar Preferences Number of Married Couples
All four
Three
Two
One
None 35
122
110
69
39
Suppose that a married couple is selected at random.

(a) Use the data to estimate the probability that they will have 0, 1, 2, 3, or 4 personality preferences in common. (Enter your answers to 2 decimal places.)

0 1 2 3 4

(b) Do the probabilities add up to 1? Why should they?



(c)What is the sample space in this problem?

a.0, 1, 2, 3 personality preferences in common

b.1, 2, 3, 4 personality preferences in common

c.0, 1, 2, 3, 4, 5 personality preferences in common

d.0, 1, 2, 3, 4 personality preferences in common

Answer 1

(a) Shown in table.

(b) All the probabilities add up to be 1.

(c) Sample space is, S = {0, 1, 2, 3, 4}

Explanation:

Let X = number of similar preference.

The formula to compute the probability of an event E is,

`P(E)=(n(E))/(N)`

Here,

n (E) = favorable outcomes of event E

N = total number of outcomes.

(a)

Compute the probability that a randomly selected married couple has 0 similar preference as follows:

`P(X = 0)=(39)/(375)=0.104`

Compute the probability that a randomly selected married couple has 1 similar preference as follows:

`P(X = 1)=(69)/(375)=0.184`

Compute the probability that a randomly selected married couple has 2 similar preference as follows:

`P(X = 2)=(110)/(375)=0.293`

Compute the probability that a randomly selected married couple has 3 similar preference as follows:

`P(X = 3)=(122)/(375)=0.325`

Compute the probability that a randomly selected married couple has 4 similar preference as follows:

`P(X = 2)=(35)/(375)=0.093`

(b)

Compute the sum of all probabilities as follows:

`\sum P (X=x)=P (X=0)+P (X=1)+P (X=2)+P (X=3)+P (X=4)\\=0.104+0.184+0.293+0.325+0.093\\=0.999\\\approx1`

All the probabilities add up to be 1.

According to the probability distribution, all the probabilities of every sample space value must add up to 1.

(c)

The sample space in this case be defined as the number of similar preference married couples have.

S = {0, 1, 2, 3, 4}