Question

A group of people were given a personality test to determine if they were Type A or Type B. The results are shown in the table below:

Male Female
Type A 55 75
Type B 48 22


Compare P(Male or Type B) with P(Male | Type B).
P(Male or Type B) > P(Male | Type B)
P(Male or Type B) = P(Male | Type B)
P(Male or Type B) < P(Male | Type B)
There is not enough information.

Answer 1

After calculating the probabilities, P(Male or Type B) is 62.5%, and P(Male | Type B) is 68.57%. Therefore, P(Male or Type B) is less than P(Male | Type B).

Step-by-step explanation:

To compare P(Male or Type B) with P(Male | Type B), we first need to calculate the probabilities. From the table provided, there are:

• 55 male Type A individuals.
• 75 female Type A individuals.
• 48 male Type B individuals.
• 22 female Type B individuals.

Total number of males is 55 (Type A) + 48 (Type B) = 103 males.
Total number of individuals with Type B personality is 48 (male) + 22 (female) = 70 people.
Therefore, P(Male or Type B) is the probability of being either male or having a Type B personality. This is calculated by adding the probabilities of the individual categories and then subtracting the intersection (male and Type B) since it is counted in both categories:

P(Male) + P(Type B) - P(Male and Type B) = (103/200) + (70/200) - (48/200) = 0.515 + 0.350 - 0.240 = 0.625 or 62.5%.

Now, P(Male | Type B) is the conditional probability of being male given that the individual has a Type B personality:

P(Male | Type B) = P(Male and Type B) / P(Type B) = (48/200) / (70/200) = 0.6857 or 68.57%.

Since 62.5% < 68.57%, we can conclude that P(Male or Type B) < P(Male | Type B).