Six coins are tossed simultaneously. The probability that at least one tail turns up is

Options

A. `63/64`
B. `1/64`
C. `3/32`
D. None of these

Answer

To solve the problem of finding the probability that at least one tail turns up when six coins are tossed simultaneously, we can follow these steps:

### Step 1: Understand the total outcomes
When a single coin is tossed, there are 2 possible outcomes: Heads (H) or Tails (T). Therefore, when 6 coins are tossed, the total number of outcomes can be calculated as:
\[ \text{Total outcomes} = 2^6 = 64 \]

### Step 2: Calculate the probability of getting no tails
The only way to get no tails is to have all coins show heads. The probability of getting heads on a single coin toss is \( \frac{1}{2} \). Thus, the probability of getting heads on all 6 coins is:
\[ P(\text{No tails}) = \left(\frac{1}{2}\right)^6 = \frac{1}{64} \]

### Step 3: Calculate the probability of getting at least one tail
To find the probability of getting at least one tail, we can use the complement rule. The complement of getting at least one tail is getting no tails. Therefore, we can calculate the probability of getting at least one tail as:
\[ P(\text{At least one tail}) = 1 - P(\text{No tails}) = 1 - \frac{1}{64} \]

### Step 4: Simplify the expression
Now, we simplify the expression:
\[ P(\text{At least one tail}) = 1 - \frac{1}{64} = \frac{64}{64} - \frac{1}{64} = \frac{63}{64} \]

### Final Answer
Thus, the probability that at least one tail turns up when six coins are tossed is:
\[ \frac{63}{64} \]

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