A fair coin is tossed and an unbiased dice rolled together. What is the probability of getting a 2 or 4 or 6 along with head?

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. `(1)/(4)`
D. `(1)/(6)`

Answer

To solve the problem of finding the probability of getting a 2, 4, or 6 along with heads when a fair coin is tossed and an unbiased die is rolled, we can follow these steps:

### Step 1: Identify the Sample Space
When a fair coin is tossed, there are 2 possible outcomes: Heads (H) and Tails (T). When an unbiased die is rolled, there are 6 possible outcomes: 1, 2, 3, 4, 5, and 6.

To find the total sample space when both the coin and the die are considered together, we can list all the combinations of the outcomes:

- H with 1
- H with 2
- H with 3
- H with 4
- H with 5
- H with 6
- T with 1
- T with 2
- T with 3
- T with 4
- T with 5
- T with 6

Thus, the total number of outcomes in the sample space is:
\[ \text{Total Outcomes} = 2 \text{ (for coin)} \times 6 \text{ (for die)} = 12 \]

### Step 2: Identify the Favorable Outcomes
We need to find the favorable outcomes where the die shows either a 2, 4, or 6 along with heads. The favorable outcomes are:

- H with 2
- H with 4
- H with 6

Thus, the number of favorable outcomes is:
\[ \text{Favorable Outcomes} = 3 \]

### Step 3: Calculate the Probability
The probability \( P \) of an event is given by the formula:
\[ P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} \]

Substituting the values we found:
\[ P(\text{Getting 2, 4, or 6 with Heads}) = \frac{3}{12} = \frac{1}{4} \]

### Final Answer
The probability of getting a 2, 4, or 6 along with heads is \( \frac{1}{4} \).

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