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You roll a $6$ -sided die two times. $\newline$ What is the probability of rolling a number greater than $3$ and then rolling an odd number? $\newline$ 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

3

and then rolling an odd number?

\newline

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

\newline

\_\_\_\_\_

Identify total possible outcomes: Identify the total number of possible outcomes for each roll of the die. Since the die is $6$ -sided, there are $6$ possible outcomes for each roll.
Determine favorable outcomes ( $1$ st event): Determine the number of favorable outcomes for the first event, which is rolling a number greater than $3$ . The numbers greater than $3$ on a $6$ -sided die are $4$ , $5$ , and $6$ . Therefore, there are $3$ favorable outcomes for the first event.
Determine favorable outcomes ( $2$ nd event): Determine the number of favorable outcomes for the second event, which is rolling an odd number. The odd numbers on a $6$ -sided die are $1$ , $3$ , and $5$ . Therefore, there are $3$ favorable outcomes for the second event.
Calculate combined probability: Since the two rolls are independent events, the probability of both events occurring is the product of their individual probabilities. Calculate the probability of the first event (rolling a number greater than $3$ ) and the second event (rolling an odd number).
Calculate probability ( $1$ st event): The probability of rolling a number greater than $3$ is $3$ favorable outcomes out of $6$ possible outcomes, which simplifies to $\frac{1}{2}$ . The probability of rolling an odd number is also $3$ favorable outcomes out of $6$ possible outcomes, which simplifies to $\frac{1}{2}$ .
Calculate probability ( $2$ nd event): Multiply the probabilities of the two independent events to find the combined probability. The probability of rolling a number greater than $3$ and then rolling an odd number is $(\frac{1}{2}) \times (\frac{1}{2})$ .
Multiply probabilities: Perform the multiplication to find the final probability. $(\frac{1}{2}) \times (\frac{1}{2}) = \frac{1}{4}$ .

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