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You roll a -sided die two times.What is the probability of rolling a number less than and then rolling a number greater than ?Simplify your answer and write it as a fraction or whole number._____
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Q. You roll a -sided die two times.What is the probability of rolling a number less than and then rolling a number greater than ?Simplify your answer and write it as a fraction or whole number._____
- Probability of Rolling < : The possible outcomes of rolling a die are . The probability of rolling a number less than is P(\text{Rolling a number} < 6) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}. Favorable outcomes for rolling a number less than are , which gives us favorable outcomes. Total outcomes are because there are sides on the die. So, P(\text{Rolling a number} < 6) = \frac{5}{6}.
- Probability of Rolling > : Next, we need to find the probability of rolling a number greater than . The only number greater than on a -sided die is .So, the probability of rolling a number greater than is P(\text{Rolling a number} > 5) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}.Favorable outcomes for rolling a number greater than are \{\}, which gives us favorable outcome.Total outcomes remain .So, P(\text{Rolling a number} > 5) = \frac{1}{6}.
- Combined Probability Calculation: Now, we need to find the combined probability of both events happening in sequence. The probability of rolling a number less than and then rolling a number greater than is the product of the individual probabilities.So, P(\text{Rolling a number} < 6 \text{ and then Rolling a number} > 5) = P(\text{Rolling a number} < 6) \times P(\text{Rolling a number} > 5).P(\text{Rolling a number} < 6 \text{ and then Rolling a number} > 5) = \frac{5}{6} \times \frac{1}{6}.
- Combined Probability Calculation: Now, we need to find the combined probability of both events happening in sequence. The probability of rolling a number less than and then rolling a number greater than is the product of the individual probabilities.So, P(\text{Rolling a number} < 6 \text{ and then Rolling a number} > 5) = P(\text{Rolling a number} < 6) \times P(\text{Rolling a number} > 5).P(\text{Rolling a number} < 6 \text{ and then Rolling a number} > 5) = (\frac{5}{6}) \times (\frac{1}{6}).Performing the multiplication, we get:P(\text{Rolling a number} < 6 \text{ and then Rolling a number} > 5) = (\frac{5}{6}) \times (\frac{1}{6}) = \frac{5}{36}.This is the simplified fraction that represents the probability of the two events happening in sequence.
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