1. Consider the two independent rolls of a fair 6 -sided dice. What is the probability of getting at least one 6 given the two rolls land on different numbers?

Question

UpStudy AI · Accepted Answer

To determine the probability of getting **at least one 6** given that the **two rolls land on different numbers**, we'll use conditional probability.

### Definitions:
- **Event A**: At least one of the two rolls is a 6.
- **Event B**: The two rolls land on different numbers.

We need to compute $$ P(A \mid B) $$, which is the probability of **Event A** occurring given that **Event B** has occurred. The formula for conditional probability is:

$$
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
$$

### Step 1: Calculate $$ P(B) $$ - Probability that the two rolls are different.
- **Total possible outcomes** when rolling two 6-sided dice: $$ 6 \times 6 = 36 $$.
- **Favorable outcomes for different numbers**: Total outcomes minus the outcomes where both dice show the same number. There are 6 outcomes where both dice show the same number (e.g., (1,1), (2,2), ..., (6,6)).
  
$$
P(B) = \frac{36 - 6}{36} = \frac{30}{36} = \frac{5}{6}
$$

### Step 2: Calculate $$ P(A \cap B) $$ - Probability that at least one 6 occurs and the two rolls are different.
- **Cases where at least one 6 occurs and the numbers are different**:
  1. First die is 6 and the second die is not 6: $$ 5 $$ outcomes.
  2. Second die is 6 and the first die is not 6: $$ 5 $$ outcomes.
  
$$
P(A \cap B) = \frac{5 + 5}{36} = \frac{10}{36} = \frac{5}{18}
$$

### Step 3: Compute the Conditional Probability $$ P(A \mid B) $$.

$$
P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{5}{18}}{\frac{5}{6}} = \frac{5}{18} \times \frac{6}{5} = \frac{1}{3}
$$

### Final Answer:
The probability is **1⁄3**.
The probability of getting at least one 6, given that the two rolls are different, is 1⁄3.

Consider the two independent rolls of a fair 6 -sided dice. What is the
probability of getting at least one 6 given the two rolls land on different
numbers?

Upstudy AI Solution

Answer

Solution

Definitions:

Step 1: Calculate - Probability that the two rolls are different.

Step 2: Calculate - Probability that at least one 6 occurs and the two rolls are different.

Step 3: Compute the Conditional Probability .

Final Answer:

Bonus Knowledge

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