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

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

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Probability of independent and dependent events

Full solution

Q. If a fair die is rolled

3

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

2

twos?

\newline

Answer:

Determine total outcomes: Determine the total number of outcomes for rolling a die $3$ times. $\newline$ Since a die has $6$ sides, 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 = $6^3 = 216$ .
Number of ways for $2$ twos: Determine the number of ways to get exactly $2$ twos in $3$ rolls. $\newline$ We can get two twos in the following sequences: $(2,2,\text{not } 2)$ , $(2,\text{not } 2,2)$ , $(\text{not } 2,2,2)$ . $\newline$ For each sequence, there are $5$ possibilities for the 'not $2$ ' roll (since it can be any number from $1$ to $6$ except $2$ ). $\newline$ Number of ways to get exactly $2$ twos = $3$ sequences $3$ $1$ $5$ possibilities = $3$ $3$ .
Calculate probability: Calculate the probability of getting exactly $2$ twos. $\newline$ Probability = Number of ways to get exactly $2$ twos / Total number of outcomes. $\newline$ Probability = $\frac{15}{216}$ .
Simplify and round: Simplify the probability and round to the nearest thousandth. $\newline$ Probability $\approx \frac{15}{216} \approx 0.0694444444$ . $\newline$ Rounded to the nearest thousandth, the probability is $0.069$ .

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