Two coins and a die are tossed. The probability that both coins fall heads and the die shows a 3 or 6, is

Options

A. `1//8`
B. `1//12`
C. `1//16`
D. none of these

Answer

To find the probability that both coins fall heads and the die shows a 3 or 6, we can follow these steps:

### Step 1: Calculate the probability of both coins showing heads.
When tossing a coin, the probability of getting heads (H) is \( \frac{1}{2} \). Since we have two coins, we need to find the probability that both coins show heads.

The probability of both coins showing heads is calculated as:
\[ P(\text{both heads}) = P(H) \times P(H) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]

### Step 2: Calculate the probability of the die showing a 3 or 6.
A die has 6 faces, numbered from 1 to 6. The favorable outcomes for our requirement (showing a 3 or 6) are 2 outcomes: {3, 6}.

The probability of the die showing a 3 or 6 is:
\[ P(\text{3 or 6}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{6} = \frac{1}{3} \]

### Step 3: Calculate the combined probability of both events.
Since the events of tossing the coins and rolling the die are independent, we can find the combined probability by multiplying the probabilities of the two independent events:
\[ P(\text{both heads and die shows 3 or 6}) = P(\text{both heads}) \times P(\text{3 or 6}) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12} \]

### Final Answer:
Thus, the probability that both coins fall heads and the die shows a 3 or 6 is \( \frac{1}{12} \).

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