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You roll a $6$ -sided die two times. $\newline$ What is the probability of rolling a number less than $6$ and then rolling a number greater than $5$ ? $\newline$ Simplify your answer and write it as a fraction or whole number. $\newline$ _____

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Probability of independent and dependent events

Full solution

Q. You roll a

6

-sided die two times.

\newline

What is the probability of rolling a number less than

6

and then rolling a number greater than

5

?

\newline

Simplify your answer and write it as a fraction or whole number.

\newline

_____

Probability of Rolling < $6$ : The possible outcomes of rolling a die are $\{1, 2, 3, 4, 5, 6\}$ . The probability of rolling a number less than $6$ is $P(\text{Rolling a number} < 6) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}$ . Favorable outcomes for rolling a number less than $6$ are $\{1, 2, 3, 4, 5\}$ , which gives us $5$ favorable outcomes. Total outcomes are $6$ because there are $6$ sides on the die. So, $P(\text{Rolling a number} < 6) = \frac{5}{6}$ .
Probability of Rolling > $5$ : Next, we need to find the probability of rolling a number greater than $5$ . The only number greater than $5$ on a $6$ -sided die is $6$ . $\newline$ So, the probability of rolling a number greater than $5$ is $P(\text{Rolling a number} > 5) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}$ . $\newline$ Favorable outcomes for rolling a number greater than $5$ are \{ $6$ \}, which gives us $1$ favorable outcome. $\newline$ Total outcomes remain $6$ . $\newline$ 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 $6$ and then rolling a number greater than $5$ is the product of the individual probabilities. $\newline$ So, $P(\text{Rolling a number} < 6 \text{ and then Rolling a number} > 5) = P(\text{Rolling a number} < 6) \times P(\text{Rolling a number} > 5)$ . $\newline$ $P(\text{Rolling a number} < 6 \text{ and then Rolling a number} > 5) = \frac{5}{6} \times \frac{1}{6}$ .
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 $6$ and then rolling a number greater than $5$ is the product of the individual probabilities. $\newline$ So, $P(\text{Rolling a number} < 6 \text{ and then Rolling a number} > 5) = P(\text{Rolling a number} < 6) \times P(\text{Rolling a number} > 5)$ . $\newline$ $P(\text{Rolling a number} < 6 \text{ and then Rolling a number} > 5) = (\frac{5}{6}) \times (\frac{1}{6})$ .Performing the multiplication, we get: $\newline$ $P(\text{Rolling a number} < 6 \text{ and then Rolling a number} > 5) = (\frac{5}{6}) \times (\frac{1}{6}) = \frac{5}{36}$ . $\newline$ This is the simplified fraction that represents the probability of the two events happening in sequence.

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