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You roll a $6$ -sided die two times. $\newline$ What is the probability of rolling a number greater than $4$ and then rolling a $4$ ? $\newline$ 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 greater than

4

and then rolling a

4

?

\newline

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

\newline

_____

Probability of Rolling > $4$ : The possible outcomes of rolling a $6$ -sided die are $\{1, 2, 3, 4, 5, 6\}$ . The probability of rolling a number greater than $4$ , which means rolling either a $5$ or a $6$ , is $P(\text{Rolling a number} > 4) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}.$ $= \frac{2}{6}$ $= \frac{1}{3}$
Probability of Rolling $4$ : The probability of rolling a $4$ on the second roll is $P(\text{Rolling a 4}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}.$ $= \frac{1}{6}$
Probability of Sequential Events: The probability of both events happening in sequence (rolling a number greater than $4$ and then rolling a $4$ ) is the product of their individual probabilities. This is because the rolls are independent events. $\newline$ $P(\text{Rolling a number} > 4 \text{ and then a } 4) = P(\text{Rolling a number} > 4) \times P(\text{Rolling a } 4)$ $\newline$ $= \frac{1}{3} \times \frac{1}{6}$ $\newline$ $= \frac{1}{18}$

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\newline

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\newline

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\newline

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