Four coins are tossed once .Find the probability of exactly 3 tails.

Options

A. `(1)/(16)`
B. `(1)/(4)`
C. `(5)/(16)`
D. `(4)/(3)`

Answer

To find the probability of getting exactly 3 tails when tossing 4 coins, we can follow these steps:

### Step 1: Understand the total outcomes
When tossing 4 coins, each coin has 2 possible outcomes: heads (H) or tails (T). Therefore, the total number of outcomes when tossing 4 coins is calculated as:
\[ \text{Total outcomes} = 2^n \]
where \( n \) is the number of coins. Here, \( n = 4 \):
\[ \text{Total outcomes} = 2^4 = 16 \]

### Step 2: Determine the favorable outcomes
Next, we need to find the number of favorable outcomes where exactly 3 coins show tails. To do this, we can choose 3 coins to be tails out of the 4 coins, and the remaining coin will be heads. The number of ways to choose 3 coins from 4 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 4 \) and \( r = 3 \):
\[ \binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4 \times 3!}{3! \times 1!} = 4 \]
So, there are 4 favorable outcomes for getting exactly 3 tails.

### Step 3: Calculate the probability
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P(\text{exactly 3 tails}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{4}{16} = \frac{1}{4} \]

### Final Answer
Thus, the probability of getting exactly 3 tails when tossing 4 coins is:
\[ \frac{1}{4} \]

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