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.
Answer
Author: Mohammad-Ali Bandzar| Date:Oct 16 2020
(a) What is the distribution of the number of times this event occurs in 20 rolls?
This question is asking for the binomial probability as all our trials are independent.
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
y represents the desired number of successful outcomes

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 $n=20$
$Y$ ~ $Binomial(n, \theta )$
$Y$ ~ $Binomial(20, 2/3)$
(b) Calculate the probability that the event occurs five times
From part a above, we know: $$P(Y=y)={n\choose y}\theta ^y(1-\theta)^{n-y}$$ we will plug in: $n=20$, $\theta=2/3$, $y=5$ $$P(Y=5)={20\choose 5}(\frac{2}{3}) ^5(1-\frac{2}{3})^{20-5}$$ $$P(Y=5)=15504*(\frac{32}{243})(\frac{1}{14348907})$$ $$P(Y=5)=\frac{165376}{1162261467}$$ $$P(Y=5)\approx 0.000$$