100 students appeared for two examinations .60 passed the first , 50 passed the second and 30 passed both . Find the probability that a student selected at random has failed in both examinations .

Answer

To solve the problem step by step, we will follow the logical reasoning outlined in the video transcript.

### Step 1: Define the given data
- Total number of students = 100
- Students who passed the first examination (P(F)) = 60
- Students who passed the second examination (P(S)) = 50
- Students who passed both examinations (P(F ∩ S)) = 30

### Step 2: Calculate the probabilities
1. Probability of passing the first examination:
\[ P(F) = \frac{60}{100} = 0.6 \]

2. Probability of passing the second examination:
\[ P(S) = \frac{50}{100} = 0.5 \]

3. Probability of passing both examinations:
\[ P(F ∩ S) = \frac{30}{100} = 0.3 \]

### Step 3: Calculate the probability of passing at least one examination
To find the probability of passing at least one examination (P(F ∪ S)), we can use the formula:
\[ P(F ∪ S) = P(F) + P(S) - P(F ∩ S) \]
Substituting the values we calculated:
\[ P(F ∪ S) = 0.6 + 0.5 - 0.3 \]
\[ P(F ∪ S) = 0.8 \]

### Step 4: Calculate the probability of failing both examinations
The probability of failing both examinations is the complement of passing at least one examination:
\[ P(\text{Failing both}) = 1 - P(F ∪ S) \]
Substituting the value we found:
\[ P(\text{Failing both}) = 1 - 0.8 = 0.2 \]

### Step 5: Conclusion
Thus, the probability that a student selected at random has failed in both examinations is:
\[ \boxed{0.2} \]

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