Five coins are tossed simultaneously. The probability that at least on head turning up, is

Options

A. `1//32`
B. `5//32`
C. `7//16`
D. `31//32`

Explanation

To solve the problem of finding the probability that at least one head appears when tossing five coins simultaneously, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Total Outcomes**:
When tossing a coin, there are two possible outcomes: heads (H) or tails (T). For five coins, the total number of outcomes is calculated as:
\[ \text{Total outcomes} = 2^5 = 32 \]

2. **Identify the Complement Event**:
Instead of directly calculating the probability of getting at least one head, we can find the probability of the complementary event, which is getting no heads at all (i.e., all tails).

3. **Calculate the Probability of All Tails**:
The probability of getting tails on a single toss of a coin is:
\[ P(T) = \frac{1}{2} \]
Therefore, the probability of getting tails on all five tosses (all tails) is:
\[ P(\text{All Tails}) = P(T) \times P(T) \times P(T) \times P(T) \times P(T) = \left(\frac{1}{2}\right)^5 = \frac{1}{32} \]

4. **Use the Complement Rule**:
The probability of getting at least one head is given by the complement of the probability of getting all tails:
\[ P(\text{At least one head}) = 1 - P(\text{All Tails}) = 1 - \frac{1}{32} \]

5. **Calculate the Final Probability**:
Now, we can calculate:
\[ P(\text{At least one head}) = 1 - \frac{1}{32} = \frac{32}{32} - \frac{1}{32} = \frac{31}{32} \]

Thus, the probability that at least one head turns up when tossing five coins is:
\[ \frac{31}{32} \]

### Final Answer:
The probability that at least one head turns up is \(\frac{31}{32}\).

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