Answer :
Sure, I'd be happy to help you solve this step by step.
### Step-by-Step Solution
Given the table:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline & \text{Chemistry} & \text{Biology} & \text{Physics} & \text{Total} \\ \hline \text{Male} & 12 & 14 & 16 & 42 \\ \hline \text{Female} & 16 & 20 & 12 & 48 \\ \hline \text{Total} & 28 & 34 & 28 & 90 \\ \hline \end{array} \][/tex]
#### 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 will use the principle of inclusion-exclusion.
1. Number of males:
Total males = 42
2. Number of people who study Biology:
Total Biology students = 34
3. Number of males who study Biology:
Males who study Biology = 14
Using the principle of inclusion-exclusion, the number of people who are male or study Biology or both is given by:
[tex]\[ \text{Total} = (\text{Total males}) + (\text{Total Biology students}) - (\text{Males who study Biology}) \][/tex]
Substituting the numbers:
[tex]\[ \text{Total} = 42 + 34 - 14 = 62 \][/tex]
So, the number of people who are male OR study Biology OR both is 62.
#### b) What is the probability that any person is male OR studies Biology OR both?
We need to find the probability that a randomly selected person fits into one of these categories. This probability is calculated by dividing the number of favorable outcomes by the total number of people.
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of people}} = \frac{62}{90} \][/tex]
To simplify the fraction:
[tex]\[ \frac{62}{90} \approx 0.6889 \][/tex]
So, the probability that any person is male OR studies Biology OR both is approximately 0.6889.
### Summary
1. Number of people who are male OR study Biology OR both: 62
2. Probability that any person is male OR studies Biology OR both: 0.6889
### Step-by-Step Solution
Given the table:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline & \text{Chemistry} & \text{Biology} & \text{Physics} & \text{Total} \\ \hline \text{Male} & 12 & 14 & 16 & 42 \\ \hline \text{Female} & 16 & 20 & 12 & 48 \\ \hline \text{Total} & 28 & 34 & 28 & 90 \\ \hline \end{array} \][/tex]
#### 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 will use the principle of inclusion-exclusion.
1. Number of males:
Total males = 42
2. Number of people who study Biology:
Total Biology students = 34
3. Number of males who study Biology:
Males who study Biology = 14
Using the principle of inclusion-exclusion, the number of people who are male or study Biology or both is given by:
[tex]\[ \text{Total} = (\text{Total males}) + (\text{Total Biology students}) - (\text{Males who study Biology}) \][/tex]
Substituting the numbers:
[tex]\[ \text{Total} = 42 + 34 - 14 = 62 \][/tex]
So, the number of people who are male OR study Biology OR both is 62.
#### b) What is the probability that any person is male OR studies Biology OR both?
We need to find the probability that a randomly selected person fits into one of these categories. This probability is calculated by dividing the number of favorable outcomes by the total number of people.
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of people}} = \frac{62}{90} \][/tex]
To simplify the fraction:
[tex]\[ \frac{62}{90} \approx 0.6889 \][/tex]
So, the probability that any person is male OR studies Biology OR both is approximately 0.6889.
### Summary
1. Number of people who are male OR study Biology OR both: 62
2. Probability that any person is male OR studies Biology OR both: 0.6889
