Question

A manager of a store wants to give away some money to her well-earned employees. So, she sets up 20 vases with 1(worthless) lead coin and 1 gold coin in each. Then, she lets an employee roll a single 20-sided die to determine how many vases the employee can sample from. If the employee rolls a k on the die, they can draw 1 coin from k-many vases. (For example, if the employee rolls a 5, they walk away with 5 coins, some of which are worthless lead, but the rest are gold). The die is fair, vases are indistinguishable, and a gold coin feels exactly like a lead coin, so everything here is perfectly random.

a) What is the probability that the employee walks away with EXACTLY 8 gold coins?

b) What is the probability that the employee walks away with AT LEAST 8 gold coins?

c) What is the expected value of the number of gold coins a person wins?

Answer 1

Let X be random variable denoting no. which is rolled by employee, then 0 otherwise y = random. variable no. of gold coins then YĪX=x ~ Binomial(x, 1/2). 0,1,..., x r1,2, ..., 20 a. Probability that the employee walks away with eractly 8 gold coins -P(Y-8) = Σ (i ) (1/2)"(1/20) = 0.0808

R code:

x=8:20
p=choose(x,8)*(0.5)^x*(1/20)
round(sum(p))

b. Probability that the employee walks away with at least 8 gold coins = 0.2652

R code:

sum=0
for(x in 8:20)
{
for(y in 8:x)
{
sum=sum+choose(x,y)*(0.5)^x*(1/20)
}
}
sum

c. Probability that the employee walks away with exactly t gold coins 20 Ply = 0) = 0.0500. P(У = 1) = 0.0100. P(y = 2) = 0.0100. P(Y= 3)= 0.0999. P(Y-4)= 0.0906, P(Y= 5) = 0.0987, P(y = 6) = 0.0961. P(У = 7) = 0.0905, P(y-8) = 0.0808. P(y = 9) = 0.0668. P(y-10) = 0.05, P(y-11) = 0.0332. P(y= 12)= 0.0192. P(y-13)= 0.0095, P(yー14)= 0.0039. P(y-15)= 0.0013. P(y-16)= 0.0004. P(y-17)=0.00007, P(Y= 18)= 0.00001, P(Y-19)= 0.000001, P(Y-20)= 0.00000005 E(Y) = 0 * 0.0500 1 * 0.0100 2 * 0.0100 3 * 0.0999 + 4 * 0.0996 5 * 0.0987+ 60.0961 + 7*0.0905 8 0.0808 90.0668+100.05 +11* 0.0332 +120.0192 13* 0.0095 14 0.0039+15 0.0013 +160.0004 + 17 * 0.00007 + 18 * 0.00001 19 * 0.000001 + 20 * 0.00000005-4.98029 ะ 5