Let's analyze the probability of rolling a 4 or doubles when two dice are rolled.

Step 1: Define the Outcomes

When two six-sided dice are rolled, the total number of possible outcomes is: $6 \times 6 = 36$

Step 2: Rolling a Sum of 4

We count the number of ways to get a sum of 4. The possible combinations for a sum of 4 are:

1. (1, 3)
2. (2, 2)
3. (3, 1)

Thus, there are 3 outcomes where the sum is 4.

Step 3: Rolling Doubles

We also count the number of ways to get doubles, which means the numbers on both dice are the same. The possible doubles are:

1. (1, 1)
2. (2, 2)
3. (3, 3)
4. (4, 4)
5. (5, 5)
6. (6, 6)

Thus, there are 6 outcomes where we roll doubles.

Step 4: Overlap Between Rolling a 4 and Doubles

Now, we look for outcomes that satisfy both conditions (rolling a 4 and rolling doubles). The only outcome that meets both criteria is (2, 2). This means there is 1 outcome in common.

Step 5: Use the Inclusion-Exclusion Principle

We calculate the probability of rolling a 4 or doubles using the formula: $P(\text{rolling a 4 or doubles}) = P(\text{rolling a 4}) + P(\text{rolling doubles}) - P(\text{both})$

• $P(\text{rolling a 4}) = \frac{3}{36}$
• $P(\text{rolling doubles}) = \frac{6}{36}$
• $P(\text{both}) = \frac{1}{36}$

$P(\text{rolling a 4 or doubles}) = \frac{3}{36} + \frac{6}{36} - \frac{1}{36} = \frac{8}{36} = \frac{2}{9}$

Final Answer

The probability of rolling a 4 or doubles is: $\frac{2}{9}$

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Do you need a detailed explanation or have any questions?

Related Questions:

1. What is the probability of rolling a sum of 7 with two dice?
2. What is the probability of getting doubles when rolling two dice?
3. How many ways can two dice roll a sum of 8?
4. What is the probability of rolling a sum of 4 or 7?
5. How does the probability change if one die is biased?

Tip:

When calculating probabilities for "either/or" scenarios, use the inclusion-exclusion principle to avoid double-counting overlapping events.