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

- What is the distribution of the number of times this event occurs in 20 rolls?
- 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: $n=20$, $\theta=2/3$, $y=5$ $$P(Y=5)={20\choose 5}(\frac{2}{3}) ^5(1-\frac{2}{3})^{20-5}$$ $$P(Y=5)=15504*(\frac{32}{243})(\frac{1}{14348907})$$ $$P(Y=5)=\frac{165376}{1162261467}$$ $$P(Y=5)\approx 0.000$$