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

5

?

\newline

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

\newline

_____

Determine total possible outcomes: Determine the total number of possible outcomes for each roll of the die. $\newline$ Since the die is $6$ -sided, there are $6$ possible outcomes for each roll.
Find favorable outcomes ( $1$ st event): Determine the number of favorable outcomes for the first event (rolling an odd number). $\newline$ The odd numbers on a $6$ -sided die are $1$ , $3$ , and $5$ . So there are $3$ favorable outcomes for rolling an odd number.
Find favorable outcomes ( $2$ nd event): Determine the number of favorable outcomes for the second event (rolling a number less than $5$ ). $\newline$ The numbers less than $5$ on a $6$ -sided die are $1$ , $2$ , $3$ , and $4$ . So there are $4$ favorable outcomes for rolling a number less than $5$ .
Calculate probability of both events: Calculate the probability of both events happening in sequence. $\newline$ Since the two rolls are independent events, the probability of both occurring is the product of their individual probabilities. $\newline$ The probability of rolling an odd number on the first roll is $\frac{3}{6}$ (or $\frac{1}{2}$ after simplifying). $\newline$ The probability of rolling a number less than $5$ on the second roll is $\frac{4}{6}$ (or $\frac{2}{3}$ after simplifying).
Multiply probabilities for combined probability: Multiply the probabilities of the two independent events to find the combined probability. $\newline$ Probability of rolling an odd number and then a number less than $5 = \left(\frac{1}{2}\right) \times \left(\frac{2}{3}\right) = \frac{1}{3}$

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\newline

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\newline

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\newline

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\newline

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\newline

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