In a class 40% of the students offered Physics 20% offered Chemistry and 5% offered both. If a student is selected at random, find the probability that he has offered Physics or Chemistry only.

Options

A. `45%`
B. `55%`
C. `36 %`
D. none of these

Answer

To solve the problem, we need to find the probability that a randomly selected student has offered Physics or Chemistry only. We can use the principle of inclusion-exclusion for this.

### Step 1: Define the probabilities
Let:
- \( P \) = Probability of a student offering Physics = 40% = \( \frac{40}{100} = 0.4 \)
- \( C \) = Probability of a student offering Chemistry = 20% = \( \frac{20}{100} = 0.2 \)
- \( P \cap C \) = Probability of a student offering both Physics and Chemistry = 5% = \( \frac{5}{100} = 0.05 \)

### Step 2: Use the formula for the union of two events
The probability of a student offering either Physics or Chemistry (or both) is given by:
\[ P(P \cup C) = P(P) + P(C) - P(P \cap C) \]

### Step 3: Substitute the values into the formula
Substituting the values we have:
\[ P(P \cup C) = 0.4 + 0.2 - 0.05 \]

### Step 4: Calculate the result
Now, we calculate:
\[ P(P \cup C) = 0.4 + 0.2 - 0.05 = 0.6 - 0.05 = 0.55 \]

### Step 5: Convert to percentage
To express this probability as a percentage:
\[ 0.55 \times 100 = 55\% \]

### Step 6: Find the probability of offering only Physics or only Chemistry
To find the probability of a student offering Physics or Chemistry only, we need to exclude those who offered both:
\[ P(\text{Physics only}) = P(P) - P(P \cap C) = 0.4 - 0.05 = 0.35 \]
\[ P(\text{Chemistry only}) = P(C) - P(P \cap C) = 0.2 - 0.05 = 0.15 \]

### Step 7: Add the probabilities of Physics only and Chemistry only
Now, we add these two probabilities:
\[ P(\text{Physics only or Chemistry only}) = P(\text{Physics only}) + P(\text{Chemistry only}) = 0.35 + 0.15 = 0.5 \]

### Step 8: Convert to percentage
Finally, convert this to percentage:
\[ 0.5 \times 100 = 50\% \]

### Final Answer
The probability that a randomly selected student has offered Physics or Chemistry only is **50%**.

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