Question: "The number of twos that appear after a..."

Osvaldo Apodaca

Answered question

2022-01-18

The number of twos that appear after a die is rolled ten times is counted. Find the two-digit standard deviation if the experiment is repeated.

Answer & Explanation

Annie Levasseur

Beginner2022-01-18Added 30 answers

Explanation: In this experiment, the die is rolled repeatedly in identical independent trials with the same requirement for "success" (rolling a "2" each time). As a result, it will have a binomial distribution, and the likelihood of rolling k 2s will be as follows:
$(\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k}$ , k=0,1,2,..., n
Where:
n is the number of trials in the experiment (10), and
p is the probability of success in each trial $(\frac{1}{6})$ .
Binomial distributions have a mean of np, and a standard deviation of $\sqrt{n p (1 - p)}$ .
Thus, the standard deviation $σ$ of this distribution is
$σ = \sqrt{n p (1 - p)} = \sqrt{10 \cdot \frac{1}{6} \cdot (1 - \frac{1}{6})}$
$= \sqrt{10 \cdot \frac{1}{6} \cdot \frac{5}{6}}$
$= \sqrt{\frac{50}{36}}$
$= \sqrt{\frac{5^{2} \cdot 2}{6^{2}}}$
$= \frac{5 \sqrt{2}}{6}$ .

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