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You roll a $6$ -sided die two times. $\newline$ What is the probability of rolling a $6$ 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

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 a $6$ : The possible outcomes of rolling a $6$ -sided die are $\{1, 2, 3, 4, 5, 6\}$ . The probability of rolling a $6$ is $P(\text{Rolling a 6}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{1}{6}$
Probability of Rolling > $5$ : Since there is only one number greater than $5$ on a $6$ -sided die, which is $6$ itself, the probability of rolling a number greater than $5$ is the same as the probability of rolling a $6$ . Therefore, $P(\text{Rolling a number} > 5) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{1}{6}$
Probability of Rolling $6$ and > $5$ : The probability of rolling a $6$ and then rolling a number greater than $5$ is the product of the two individual probabilities. This is because the two rolls are independent events. So, $P(\text{Rolling a 6 and then Rolling a number > 5}) = P(\text{Rolling a 6}) \times P(\text{Rolling a number > 5})$ $= \frac{1}{6} \times \frac{1}{6}$ $= \frac{1}{36}$

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\newline

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