Two dice are thrown. Find the probability that the sum is 8 or greater than 8, if 4 appears on the first die.

Options

A. `3/8`
B. `5/8`
C. `1/2`
D. none of these

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
We need to find the probability that the sum of two dice is 8 or greater than 8, given that the first die shows a 4.

### Step 2: Identify Possible Outcomes
When the first die shows a 4, the possible outcomes for the second die can be:
- (4, 1)
- (4, 2)
- (4, 3)
- (4, 4)
- (4, 5)
- (4, 6)

So, the total number of outcomes (A) when the first die is 4 is 6.

### Step 3: Determine Favorable Outcomes
Next, we need to find the outcomes where the sum is 8 or greater than 8:
- For (4, 1): Sum = 4 + 1 = 5 (not favorable)
- For (4, 2): Sum = 4 + 2 = 6 (not favorable)
- For (4, 3): Sum = 4 + 3 = 7 (not favorable)
- For (4, 4): Sum = 4 + 4 = 8 (favorable)
- For (4, 5): Sum = 4 + 5 = 9 (favorable)
- For (4, 6): Sum = 4 + 6 = 10 (favorable)

The favorable outcomes (B) where the sum is 8 or greater are:
- (4, 4)
- (4, 5)
- (4, 6)

Thus, the number of favorable outcomes is 3.

### Step 4: Calculate the Probability
The probability of an event is given by the formula:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]

Where:
- \( P(A \cap B) \) is the number of favorable outcomes (3).
- \( P(A) \) is the total number of outcomes when the first die is 4 (6).

So,
\[ P(B|A) = \frac{3}{6} = \frac{1}{2} \]

### Step 5: Conclusion
The probability that the sum is 8 or greater than 8, given that the first die shows a 4, is \( \frac{1}{2} \).

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