An unbiased coin is tossed until the first head appears or until four tosses are completed. whichever happens earlier. Which of the following statements is/are correct?
1. The probability that no head is observed is `(1)/(16)`
2 . The probability that the experiment ends with three tosses is `(1)/(8)`
Select the correct answer using the codes given below:

Options

A. Only 1
B. Only 2
C. Both 1 and 2
D. Neither 1 nor 2

Answer

To solve the problem, we need to analyze the situation where an unbiased coin is tossed until the first head appears or until four tosses are completed, whichever happens first. We will evaluate the two statements given in the question.

### Step 1: Calculate the probability of no head observed in four tosses.

The probability of getting a tail (no head) in a single toss of an unbiased coin is \( \frac{1}{2} \).

If we want to find the probability of getting tails in all four tosses, we multiply the probability of getting tails for each toss:

\[ P(\text{No head in 4 tosses}) = P(T) \times P(T) \times P(T) \times P(T) = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

### Step 2: Verify the first statement.

The first statement claims that the probability that no head is observed is \( \frac{1}{16} \). From our calculation, we found that:

\[ P(\text{No head}) = \frac{1}{16} \]

Thus, **Statement 1 is correct**.

### Step 3: Calculate the probability that the experiment ends with three tosses.

For the experiment to end with three tosses, we need to get tails on the first two tosses and heads on the third toss.

The probability of this sequence is:

\[ P(T) \times P(T) \times P(H) = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \]

### Step 4: Verify the second statement.

The second statement claims that the probability that the experiment ends with three tosses is \( \frac{1}{8} \). From our calculation, we found that:

\[ P(\text{Ends with 3 tosses}) = \frac{1}{8} \]

Thus, **Statement 2 is also correct**.

### Conclusion:

Both statements are correct. Therefore, the final answer is that both statements are true.