A coin is tossed thrice. The probability that exactly two heads show up is

Options

A. `(1)/(8)`
B. `(2)/(8)`
C. `(3)/(8)`
D. `(4)/(8)`

Answer

To find the probability that exactly two heads show up when a coin is tossed thrice, we can follow these steps:

### Step 1: Determine the total number of outcomes
When a coin is tossed, there are 2 possible outcomes: Heads (H) or Tails (T). If the coin is tossed three times, the total number of outcomes can be calculated using the formula for the total outcomes of independent events, which is \(2^n\), where \(n\) is the number of tosses.

\[ \text{Total outcomes} = 2^3 = 8 \]

### Step 2: List all possible outcomes
The possible outcomes when tossing the coin three times are:
1. HHH
2. HHT
3. HTH
4. HTT
5. THH
6. THT
7. TTH
8. TTT

### Step 3: Identify the favorable outcomes for exactly two heads
Next, we need to find the outcomes that have exactly two heads. From the list of outcomes, we can see that the following outcomes have exactly two heads:
1. HHT
2. HTH
3. THH

Thus, there are 3 favorable outcomes.

### Step 4: Calculate the probability
The probability of an event is given by the formula:

\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

Substituting the values we found:

\[ \text{Probability} = \frac{3}{8} \]

### Final Answer
The probability that exactly two heads show up when a coin is tossed thrice is \(\frac{3}{8}\).

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