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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 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 greater than

3

and then rolling a number less than

4

?

\newline

Write your answer as a percentage.

\newline

\_\_\_\_\_

%

Probability of Rolling > $3$ : The possible outcomes of rolling a $6$ -sided die are $\{1, 2, 3, 4, 5, 6\}$ . To find the probability of rolling a number greater than $3$ , we count the favorable outcomes ( $4$ , $5$ , $6$ ) and divide by the total number of outcomes. $P(\text{Rolling a number} > 3) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}$ $= \frac{3}{6}$ $= \frac{1}{2}$
Probability of Rolling < $4$ : Similarly, to find the probability of rolling a number less than $4$ , we count the favorable outcomes $(1, 2, 3)$ and divide by the total number of outcomes. $\newline$ $P(\text{Rolling a number} < 4) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}$ $\newline$ $= \frac{3}{6}$ $\newline$ $= \frac{1}{2}$
Probability of Sequential Events: The probability of both events happening in sequence (rolling a number greater than $3$ and then rolling a number less than $4$ ) is the product of their individual probabilities. $\newline$ $P(\text{Rolling a number} > 3 \text{ and then} < 4) = P(\text{Rolling a number} > 3) \times P(\text{Rolling a number} < 4)$ $\newline$ $= \frac{1}{2} \times \frac{1}{2}$ $\newline$ $= \frac{1}{4}$
Expressing Probability as Percentage: To express the probability as a percentage, we multiply the probability by $100$ . $\newline$ $(1 / 4) \times 100 = 25\%$ $\newline$ So, the probability percentage is $25\%$ .

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\newline

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