If a fair die is rolled $7$ times, what is the probability, to the nearest thousandth, of getting exactly $1$ two? $\newline$ Answer:

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Grade 8

Probability of independent and dependent events

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Q. If a fair die is rolled

7

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

1

two?

\newline

Answer:

Determine Probability of Two: Determine the probability of rolling a two on a single roll of a fair die. A fair die has six faces, so the probability of rolling any specific number, including a two, is $1$ out of $6$ .
Determine Probability of Not Two: Determine the probability of not rolling a two on a single roll of a fair die. Since there are $5$ other outcomes that are not a two, the probability of not rolling a two is $\frac{5}{6}$ .
Calculate Probability of One Two in Seven Rolls: Calculate the probability of rolling exactly one two in seven rolls. $\newline$ This event can happen in several different ways: the two can appear on the first roll, the second roll, and so on, up to the seventh roll. For each case, the other six rolls must not be a two.
Use Binomial Probability Formula: Use the binomial probability formula to calculate the probability of exactly one success (rolling a two) in seven trials (rolls). $\newline$ The binomial probability formula is $P(X=k) = \binom{n}{k} \cdot (p^k) \cdot ((1-p)^{(n-k)})$ , where: $\newline$ - $P(X=k)$ is the probability of $k$ successes in $n$ trials, $\newline$ - $\binom{n}{k}$ is the binomial coefficient, $\newline$ - $p$ is the probability of success on a single trial, $\newline$ - $(1-p)$ is the probability of failure on a single trial.
Plug Values into Formula: Plug the values into the binomial probability formula. $\newline$ Here, $n=7$ (number of trials), $k=1$ (number of successes), $p=\frac{1}{6}$ (probability of rolling a two), and $(1-p)=\frac{5}{6}$ (probability of not rolling a two). $\newline$ $P(X=1) = \binom{7}{1} \times \left(\frac{1}{6}\right)^1 \times \left(\frac{5}{6}\right)^{7-1}$
Calculate Binomial Coefficient: Calculate the binomial coefficient $\binom{7}{1}$ . $\binom{7}{1}$ is the number of ways to choose $1$ success (rolling a two) out of $7$ trials, which is simply $7$ .
Perform Calculations: Perform the calculations. $\newline$ $P(X=1) = 7 \times \left(\frac{1}{6}\right)^1 \times \left(\frac{5}{6}\right)^6$ $\newline$ $P(X=1) = 7 \times \left(\frac{1}{6}\right) \times \left(\frac{5}{6}\right)^6$
Calculate Exact Probability: Calculate the exact probability. $\newline$ $P(X=1) = 7 \times \left(\frac{1}{6}\right) \times \left(\frac{15625}{46656}\right)$ (since $\left(\frac{5}{6}\right)^6 = \frac{15625}{46656}$ ) $\newline$ $P(X=1) = 7 \times \left(\frac{1}{6}\right) \times \left(\frac{15625}{46656}\right)$ $\newline$ $P(X=1) = 7 \times \frac{15625}{6 \times 46656}$ $\newline$ $P(X=1) = \frac{109375}{279936}$
Simplify Fraction and Round: Simplify the fraction and round to the nearest thousandth. $P(X=1) \approx 0.391$ (rounded to three decimal places)