Question

The probability that a random gift box in Overwatch (PC game) has one of the character skins you want is .1. Suppose you get a gift box every game you play, and that you play until you have obtained 2 of these skins. a. What is the probability that you play until you have x boxes that do not have the desired prize? Write down the formula as well as the notation for the pdf. b. What is the probability that you play exactly 5 times? Show the R code. c. What is the probability that you play at most 5 times? Show the R code. d. How many boxes without the desired skins do you expect to get? Show the formula

Answer 1

a) Let p : Prob. ofsuccess = P ( Box with desired skins) =0.10.

Let X denotes number of boxes drawn before without the 2 desired skin boxes.

k= number of successes = 2

i.e. X denotes the number of failurers before the two successes.

P ( X =x) = P ( number of boxes that do not have desired prize)

The distribution of random variable X is negative binomial distribution with parameter

k=2 and p = 0.10.

X ~ NB (k=2, p= 0.10)

The p.m.f. of X is

b) total number of trials = 5

i.e. Number of failurers before the 2 successes is 3.

Required Probability = P ( X = 3)

by using R

> p=dnbinom(3,2,0.10)
> p
[1] 0.02916

P ( Exactly play 5 times) = 0.02916

c) Required Probability = P ( X <=3)

by using R

> p1=pnbinom(3,2,0.10)
> p1
[1] 0.08146

P ( Play at most 5 games) = 0.08146

d) Since X ~ NB (k =2, p= 0.10)

E(X) = k*q / p

E(X) = 2 *0.90 / 0.10

E(X) = 18

Expected number of boxes without desired skins = 18.