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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 greater than $3$ ? $\newline$ 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 greater than

3

?

\newline

Write your answer as a percentage.

\newline

\_\_\_\_\%

Determine Probability Less Than $4$ : Determine the probability of rolling a number less than $4$ on 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 calculated as the number of favorable outcomes divided by the total number of outcomes. P(\text{Rolling a number} < 4) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2}
Determine Probability Greater Than $3$ : Determine the probability of rolling a number greater than $3$ on a $6$ -sided die. The numbers greater than $3$ are $\{4, 5, 6\}$ . Therefore, the probability of rolling a number greater than $3$ is also calculated as the number of favorable outcomes divided by the total number of outcomes. P(\text{Rolling a number} > 3) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2}
Calculate Combined Probability: Calculate the combined probability of rolling a number less than $4$ first and then rolling a number greater than $3$ . $\newline$ Since the two rolls are independent events, the combined probability is the product of the individual probabilities. $\newline$ P(\text{Rolling a number} < 4 \text{ and then} > 3) = P(\text{Rolling a number} < 4) \times P(\text{Rolling a number} > 3) $\newline$ $= \frac{1}{2} \times \frac{1}{2}$ $\newline$ $= \frac{1}{4}$
Convert to Percentage: Convert the probability to a percentage. $\newline$ To express the probability as a percentage, multiply it by $100$ . $\newline$ $(1 / 4) \times 100 = 25\%$

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Question

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\newline

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\newline

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\newline

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\newline

Write your answer as a decimal rounded to the nearest thousandth.

\newline

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\newline

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