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You roll a $6$ -sided die two times. $\newline$ What is the probability of rolling a number less than $4$ and then rolling a number less than $4$ ? $\newline$ Write your answer as a percentage. $\newline$ $\_\_\_\_\%$

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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

4

and then rolling a number less than

4

?

\newline

Write your answer as a percentage.

\newline

\_\_\_\_\%

Calculate Probability Less Than $4$ : Determine the probability of rolling a number less than $4$ on a single roll of a $6$ -sided die. The possible outcomes of rolling a die are $\{1, 2, 3, 4, 5, 6\}$ . The numbers less than $4$ are $\{1, 2, 3\}$ . Therefore, the probability of rolling a number less than $4$ is $P(\text{Rolling a number} < 4) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2}$
Independent Event Probability: Since the die rolls are independent events, the probability of rolling a number less than $4$ on the first roll and then again on the second roll is the product of the individual probabilities. $\newline$ Using the probability from Step $1$ , we calculate $P(\text{Rolling a number} < 4 \text{ and then Rolling a number} < 4) = P(\text{Rolling a number} < 4) \times P(\text{Rolling a number} < 4)$ $\newline$ $= \frac{1}{2} \times \frac{1}{2}$ $\newline$ $= \frac{1}{4}$
Convert to Percentage: Convert the probability to a percentage. $\newline$ To convert a probability to a percentage, multiply it by $100$ . $\newline$ $(1 / 4) \times 100 = 25\%$

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\newline

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\newline

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