Question

one thousand raffle tickets are sold at $1 each. 3 tickets will be drawn at random finite

One thousand raffle tickets are sold at $1 each. Three tickets will be drawn at random (without replacement), and each will pay $191. Suppose you buy 5 gats. (A) Create a payoff table for 0, 1, 2, and 3 winning tickets among the 5 tickets you purchased. (If you do not have any winning tickets, you lose $5. If you have 1 winning ticket, you $186 since your initial $5 will not be returned to you, and so on.) (B) What is the expected value of the raffle to you? (A) Let X be the random variable for the amount won on a single raffle ticket. Create a payoff table for 0, 1, 2, and 3 winning tickets among the 5 tickets you purchased. * - $5 $186 $377 $668 ROOD (Type an integer or decimal rounded to eight decimal places as needed.) (B) The expected value of the raffle is $ (Round to the nearest cont as needed.)

Answer 1

Suppose X = number of winning tackets within 5 purchased tickets

n = purchased tickets = 5

Probability of winning ticket = p = 3 / 1000

(since 3 tickets are winning tickets out of 1000 tickets)

Hence, x follows Binomial distribution with n = 5, p = 3/1000

P(X=0) = ⁵C₀ * (3/1000)^0 * {1-(3/1000)} ^(5-0)

P(x=0) = 0.98508973

P(X=1) = ⁵C1 * (3/1000)^1 * {1-(3/1000)} ^(5-1)

P(x=1) = 0.01482081

P(x=2) = ⁵C2 * (3/1000)^2 * {1-(3/1000)} ^(5-2)

P(x=2) = 0.00008919

P(x=3) = ⁵C3 * (3/1000)^3 * {1-(3/1000)} ^(5-3)

P(x=3) = 0.00000020

Xi	-5	186	377	568
Pi	0.98508973	0.01482081	0.00008919	0.00000020

B.

Expected value

= E(X) = $\sum$ x. P(X=x)

= - 5 * 0.98508973 + 186*0.01482081 + 377*0.00008919 + 568*0.00000020

= - 2.14