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If a fair die is rolled 4 times, what is the probability, rounded to the nearest thousandth, of getting at most 1 two?
Answer:

If a fair die is rolled $4$ times, what is the probability, rounded to the nearest thousandth, of getting at most $1$ two? $\newline$ Answer:

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Probability of independent and dependent events

Full solution

Q. If a fair die is rolled

4

times, what is the probability, rounded to the nearest thousandth, of getting at most

1

two?

\newline

Answer:

Calculate Probability of Not Rolling Two: Determine the probability of not rolling a two on a single die roll. $\newline$ A fair die has $6$ sides, so the probability of rolling any specific number, including a two, is $\frac{1}{6}$ . Therefore, the probability of not rolling a two is $\frac{5}{6}$ .
Calculate Probability of Not Rolling Two in Four Rolls: Calculate the probability of not rolling a two in all four rolls. $\newline$ Since each roll is independent, we multiply the probability of not rolling a two for each roll. $\newline$ $(\frac{5}{6}) \times (\frac{5}{6}) \times (\frac{5}{6}) \times (\frac{5}{6}) = (\frac{5}{6})^4$
Calculate Probability of Rolling Exactly One Two: Calculate the probability of rolling exactly one two in four rolls. $\newline$ We need to consider all the different positions the two can appear in, which is $4$ (first, second, third, or fourth roll). The probability for each scenario is $(1/6)$ for rolling a two and $(5/6)$ for not rolling a two in the other three rolls. $\newline$ $4 \times (1/6) \times (5/6) \times (5/6) \times (5/6) = 4 \times (1/6) \times (5/6)^3$
Add Probabilities of No Twos and Exactly One Two: Add the probabilities of rolling no twos and exactly one two. $\newline$ This will give us the probability of rolling at most one two in four rolls. $\newline$ $(5/6)^4 + 4 \times (1/6) \times (5/6)^3$
Perform Calculations: Perform the calculations. $\newline$ $(\frac{5}{6})^4 = (\frac{625}{1296})$ $\newline$ $4 \times (\frac{1}{6}) \times (\frac{5}{6})^3 = 4 \times (\frac{1}{6}) \times (\frac{125}{216}) = (\frac{500}{1296})$ $\newline$ Adding these together gives: $\newline$ $(\frac{625}{1296}) + (\frac{500}{1296}) = (\frac{1125}{1296})$
Round Result: Round the result to the nearest thousandth. $\frac{1125}{1296} \approx 0.868$

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