A coin is tossed three times. If all the outcomes are identical, then I win. What is the chance that I lose?

Options

A. `3//8`
B. `3//4`
C. `1//4`
D. `5//8`

Answer

To solve the problem of finding the chance that I lose when a coin is tossed three times, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Total Outcomes**:
When a coin is tossed, there are two 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 \(2^n\), where \(n\) is the number of tosses.
\[ \text{Total outcomes} = 2^3 = 8 \]

2. **List All Possible Outcomes**:
The possible outcomes when tossing a coin three times are:
- HHH
- HHT
- HTH
- THH
- HTT
- THT
- TTH
- TTT

3. **Identify Winning Outcomes**:
According to the problem, I win if all outcomes are identical. The only outcomes where all tosses are identical are:
- HHH (all heads)
- TTT (all tails)
Thus, there are 2 winning outcomes.

4. **Identify Losing Outcomes**:
If I do not win, I lose. Therefore, the losing outcomes are all the outcomes that are not identical. Since there are 8 total outcomes and 2 winning outcomes, the number of losing outcomes is:
\[ \text{Losing outcomes} = \text{Total outcomes} - \text{Winning outcomes} = 8 - 2 = 6 \]

5. **Calculate the Probability of Losing**:
The probability of losing is calculated by dividing the number of losing outcomes by the total number of outcomes.
\[ \text{Probability of losing} = \frac{\text{Losing outcomes}}{\text{Total outcomes}} = \frac{6}{8} = \frac{3}{4} \]

### Final Answer:
The chance that I lose is \(\frac{3}{4}\) or 75%.

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