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If a fair die is rolled 3 times, what is the probability, to the nearest thousandth, of getting exactly 2 sixes?
Answer:

If a fair die is rolled $3$ times, what is the probability, to the nearest thousandth, of getting exactly $2$ sixes? $\newline$ Answer:

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Full solution

Q. If a fair die is rolled

3

times, what is the probability, to the nearest thousandth, of getting exactly

2

sixes?

\newline

Answer:

Identify Total Outcomes: Identify the total number of outcomes for rolling a die $3$ times. $\newline$ Since a die has $6$ faces, each roll has $6$ possible outcomes. For $3$ rolls, the total number of outcomes is $6 \times 6 \times 6$ . $\newline$ Total number of outcomes = $6^3 = 216$ .
Determine Ways for $2$ Sixes: Determine the number of ways to get exactly $2$ sixes in $3$ rolls. $\newline$ We can get $2$ sixes in the following sequences: $(6,6,X)$ , $(6,X,6)$ , $(X,6,6)$ , where $X$ is any number from $1$ to $5$ (not a six). $\newline$ For each sequence, there are $5$ possibilities for $X$ . $\newline$ Number of ways to get $2$ sixes = $3$ sequences $3$ $3$ $5$ possibilities for $X$ = $3$ $6$ .
Calculate Probability: Calculate the probability of getting exactly $2$ sixes. $\newline$ Probability = (Number of ways to get $2$ sixes) / (Total number of outcomes). $\newline$ Probability = $\frac{15}{216}.$
Simplify and Round: Simplify the probability and round to the nearest thousandth. $\newline$ Probability $\approx 0.0694$ (rounded to the nearest thousandth).

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