Let's break down the problem step by step:
### Part (a): How many people are male OR study Biology OR both?
To find the number of people who are either male, or study Biology, or both, we need to use the principle of inclusion and exclusion. This principle helps to ensure that we don't double-count the people who are both male and study Biology.
1. Number of males (A):
[tex]\[
A = 42
\][/tex]
2. Number of people who study Biology (B):
[tex]\[
B = 34
\][/tex]
3. Number of males who study Biology (A ∩ B):
[tex]\[
A ∩ B = 14
\][/tex]
Using the principle of inclusion and exclusion:
[tex]\[
A \cup B = A + B - A ∩ B
\][/tex]
Plugging in the values:
[tex]\[
A \cup B = 42 + 34 - 14 = 62
\][/tex]
So, the number of people who are male OR study Biology OR both is:
[tex]\[
62
\][/tex]
### Part (b): What is the probability that any person is male OR studies Biology OR both?
The probability of an event is given by the number of favorable outcomes divided by the total number of possible outcomes.
1. Number of favorable outcomes (people who are male OR study Biology OR both):
[tex]\[
\text{Favorable outcomes} = 62
\][/tex]
2. Total number of possible outcomes (total number of people):
[tex]\[
\text{Total outcomes} = 90
\][/tex]
The probability (P) is given by:
[tex]\[
P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{62}{90}
\][/tex]
Simplifying the fraction:
[tex]\[
P = \frac{62}{90} \approx 0.6889 \text{ (rounded to 4 decimal places)}
\][/tex]
Therefore, the probability that any person is male OR studies Biology OR both is approximately:
[tex]\[
0.6889 \text{ or } 68.89\%
\][/tex]
In summary:
- (a) The number of people who are male OR study Biology OR both is [tex]\(62\)[/tex].
- (b) The probability that any person is male OR studies Biology OR both is [tex]\( \approx 0.6889 \)[/tex] or [tex]\( 68.89\% \)[/tex].
