In a class of 60 students, 30 opted for Mathematics, 32 opted for Biology and 24 opted for both Mathematics and Biology. If one of these students is selected at random, find the probability that :
(i) The students opted for Mathematics or Biology.
(i) The students has opted neither Mathematics nor Biology.
(iii) The students has opted Mathematics but not Biology.

Answer

To solve the problem step by step, we will use the principles of set theory and probability.

### Given Data:
- Total number of students (U) = 60
- Number of students who opted for Mathematics (M) = 30
- Number of students who opted for Biology (B) = 32
- Number of students who opted for both Mathematics and Biology (M ∩ B) = 24

### Step 1: Find the number of students who opted for either Mathematics or Biology (M ∪ B).

Using the formula for the union of two sets:
\[ M \cup B = M + B - (M \cap B) \]
Substituting the values:
\[ M \cup B = 30 + 32 - 24 = 38 \]

### Step 2: Find the probability that a student opted for Mathematics or Biology.

The probability \( P(M \cup B) \) is given by:
\[ P(M \cup B) = \frac{M \cup B}{U} = \frac{38}{60} = \frac{19}{30} \]

### Step 3: Find the number of students who opted for neither Mathematics nor Biology.

The number of students who opted for neither is:
\[ U - (M \cup B) = 60 - 38 = 22 \]

### Step 4: Find the probability that a student opted for neither Mathematics nor Biology.

The probability \( P(\text{neither M nor B}) \) is given by:
\[ P(\text{neither M nor B}) = \frac{22}{60} = \frac{11}{30} \]

### Step 5: Find the number of students who opted for Mathematics but not Biology.

The number of students who opted for only Mathematics is:
\[ M - (M \cap B) = 30 - 24 = 6 \]

### Step 6: Find the probability that a student opted for Mathematics but not Biology.

The probability \( P(\text{only M}) \) is given by:
\[ P(\text{only M}) = \frac{6}{60} = \frac{1}{10} \]

### Summary of Results:
1. Probability that a student opted for Mathematics or Biology: \( \frac{19}{30} \)
2. Probability that a student opted for neither Mathematics nor Biology: \( \frac{11}{30} \)
3. Probability that a student opted for Mathematics but not Biology: \( \frac{1}{10} \)