# a) A survey of a sample of business students resulted in the following information, regarding the...

## Question:

a) A survey of a sample of business students resulted in the following information, regarding the genders of the individuals and their selected major:

male:
management: 40
marketing: 10
others: 30
total: 80

female:
management: 30
marketing: 20
others: 70
total: 120

What is the probability of selecting an individual, who is either male, or majoring in Management?

## Non-Mutually Exclusive:

Mutually exclusive events are events that can't happen at the same time. Say we toss a coin once, then we know that the coin will fall either heads or tails but can't be both. On the other hand, non-mutually exclusive events are the events that can happen at the same time or have an overlap.

The problem asks for event A, male, or event B, majoring in management. The rule for "OR" when two events, {eq}A {/eq} and {eq}B {/eq}, are non-mutually exclusive is given by:

$$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$$

The term {eq}P\left(A\cap B\right) {/eq} refers to the overlap of events {eq}A {/eq} and {eq}B {/eq}, that is, males who majored in management. And by looking at the data, we can see that there are {eq}40{/eq} of them. The total of all the individuals is {eq}80 + 120 = 200 {/eq}. Also, we can see that there are {eq}40{/eq} males and {eq}30{/eq} females who took management. Now, substituting values into the equation:

$$P\left(A\cup B\right)=\frac{80}{200}+\frac{40+30}{200}-\frac{40}{200} \\ P\left(A\cup B\right)= \frac{11}{20}$$

$$\boxed{P\left(A\cup B\right)= \frac{11}{20}}$$