Example Card: Finding the Probability of Inclusive Events

Now, what if two events can happen at the same time? This example covers our second key idea, inclusive events , where we must avoid double counting.

Example Problem What is the probability of rolling doubles or a sum of 6 using two number cubes?

Step by Step 1. First, determine if the events are inclusive. We need to list the outcomes for each event. 2. Outcomes for rolling doubles: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6). So, $\operatorname{P}(\text{doubles}) = \frac{6}{36}$. 3. Outcomes for rolling a sum of 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1). So, $\operatorname{P}(\text{sum of } 6) = \frac{5}{36}$. 4. Check for overlap. The outcome (3, 3) appears in both lists. This means the events are inclusive , and the probability of the overlap is $\operatorname{P}(\text{doubles and sum of } 6) = \frac{1}{36}$. 5. Use the formula for inclusive events, which subtracts the overlap to prevent counting it twice. $$ \operatorname{P}(A \text{ or } B) = \operatorname{P}(A) + \operatorname{P}(B) \operatorname{P}(A \text{ and } B) $$ $$ \operatorname{P}(\text{doubles or sum of 6}) = \operatorname{P}(\text{doubles}) + \operatorname{P}(\text{sum of 6}) \operatorname{P}(\text{doubles and sum of 6}) $$ $$ = \frac{6}{36} + \frac{5}{36} \frac{1}{36} $$ $$ = \frac{10}{36} = \frac{5}{18} $$.