find the probability that you will roll an even number exactly 5 times when you roll a six - sided number…

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successful trials, $p$ is the probability of success in a single - trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. For rolling a six - sided die, the probability of rolling an even number ($2$, $4$, or $6$) in a single roll is $p=\frac{3}{6}=\frac{1}{2}$, and $1-p=\frac{1}{2}$.

Step2: Analyze the first case ($n = 10,k = 5$)

First, calculate the combination $C(10,5)=\frac{10!}{5!(10 - 5)!}=\frac{10\times9\times8\times7\times6}{5\times4\times3\times2\times1}=252$. Then, $P(X = 5)=C(10,5)\times(\frac{1}{2})^{5}\times(\frac{1}{2})^{10 - 5}=252\times(\frac{1}{2})^{10}=252\times\frac{1}{1024}\approx0.246$.

Step3: Analyze the second case ($n = 20,k = 5$)

Calculate the combination $C(20,5)=\frac{20!}{5!(20 - 5)!}=\frac{20\times19\times18\times17\times16}{5\times4\times3\times2\times1}=15504$. Then, $P(X = 5)=C(20,5)\times(\frac{1}{2})^{5}\times(\frac{1}{2})^{20 - 5}=15504\times(\frac{1}{2})^{20}=15504\times\frac{1}{1048576}\approx0.015$.

Step4: Explain the difference

As the number of trials $n$ increases, the distribution of the number of successes becomes more spread out. When $n = 20$, there are many more possible numbers of even - number rolls. The probability of getting exactly 5 even - number rolls out of 20 is less likely compared to getting exactly 5 even - number rolls out of 10 because the binomial distribution with a larger $n$ has a wider range of possible values for the number of successes, and the probability mass is more evenly distributed among these values.

Answer:

When rolling a six - sided number cube 10 times, the probability of rolling an even number exactly 5 times is approximately $0.246$. When rolling it 20 times, the probability of rolling an even number exactly 5 times is approximately $0.015$. The second result is less because as the number of trials increases, the binomial distribution becomes more spread out, and the probability of a specific number of successes (in this case, 5) decreases.