Three unbiased coins are tossed. If the probability of getting at least 2 tails is p. Then the value of 8p -

Options

A. 0
B. 1
C. 3
D. 4

Answer

To solve the problem, we need to calculate the probability of getting at least 2 tails when tossing 3 unbiased coins. Let's break down the solution step by step.

### Step 1: Determine the Sample Space
When tossing 3 unbiased coins, the possible outcomes can be represented as follows:
- HHH (3 heads)
- HHT (2 heads, 1 tail)
- HTH (2 heads, 1 tail)
- THH (2 heads, 1 tail)
- HTT (1 head, 2 tails)
- THT (1 head, 2 tails)
- TTH (1 head, 2 tails)
- TTT (3 tails)

Thus, the sample space (S) consists of 8 outcomes:
\[ S = \{ HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \} \]

### Step 2: Identify Favorable Outcomes
Next, we need to find the outcomes that have at least 2 tails. The outcomes that satisfy this condition are:
- HTT (1 head, 2 tails)
- THT (1 head, 2 tails)
- TTH (1 head, 2 tails)
- TTT (3 tails)

Counting these, we find there are 4 favorable outcomes:
\[ \text{Favorable Outcomes} = \{ HTT, THT, TTH, TTT \} \]

### Step 3: Calculate the Probability
The probability \( p \) of getting at least 2 tails is given by the formula:
\[ p = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \]

Substituting the values we found:
\[ p = \frac{4}{8} = \frac{1}{2} \]

### Step 4: Calculate \( 8p \)
Now, we need to find the value of \( 8p \):
\[ 8p = 8 \times \frac{1}{2} = 4 \]

### Final Answer
Thus, the value of \( 8p \) is:
\[ \boxed{4} \]

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