Chapter 2 Section 3 Question 14
Question
Suppose that a symmetrical die is rolled 20 independent times, and each time we record whether or not the event $\{2, 3, 5, 6\}$ has occurred
1. What is the distribution of the number of times this event occurs in 20 rolls?
2. Calculate the probability that the event occurs five times.
From Example 2.3.3 in the book we know: $$p\scriptsize{Y}\normalsize{(y)=P(Y=y)={n\choose y}\theta ^y(1-\theta)^{n-y}}$$ n represents the number of trials
$\theta$ represents the probability of a success
A die has 6 possible outcomes all of equal probability, we have 4 desired outcomes in this question. Therefore $\theta=\frac{4}{6}=\frac{2}{3}$
We have 20 independant trials. Therefore $n=20$
$Y$ ~ $Binomial(n, \theta )$
$Y$ ~ $Binomial(20, 2/3)$
From part a above, we know: $$P(Y=y)={n\choose y}\theta ^y(1-\theta)^{n-y}$$ we will plug in: $n=20$, $\theta=2/3$, $y=5$ $$P(Y=5)={20\choose 5}(\frac{2}{3}) ^5(1-\frac{2}{3})^{20-5}$$ $$P(Y=5)=15504*(\frac{32}{243})(\frac{1}{14348907})$$ $$P(Y=5)=\frac{165376}{1162261467}$$ $$P(Y=5)\approx 0.000$$