Question

find the probability that you will roll an even number exactly 5 times when you: roll a six - sided number cube 10 times. p = 0.246 √ roll a six - sided number cube 20 times. p = 0.015 √ complete explain why the second result is less.

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 a six - sided number cube, 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

When $n = 10$ and $k = 5$, we 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

When $n = 20$ and $k = 5$, we 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 binomial distribution spreads out more. The probability of getting a fixed number of successes ($k = 5$) becomes less likely because there are more possible numbers of successes. When $n = 20$, there are many more ways to distribute the successes among the trials compared to when $n = 10$. So, the probability of getting exactly 5 successes is lower when $n = 20$ compared to when $n = 10$.

Answer:

As the number of trials ($n$) increases from 10 to 20, the binomial distribution spreads out more. There are more possible numbers of successes in 20 trials than in 10 trials. So, the probability of getting exactly 5 even - numbered rolls is lower when rolling the cube 20 times compared to 10 times.