Question

In the lottery of a certain state, players pick six different integers between 1 and 49, the order of the selection being irrelevant. The lottery commission then selects six of these numbers at random as the winning numbers. A player wins the grand prize of $1,200,000 if all six numbers that he has selected match the winning numbers. He wins the second and third prizes of $800 and $35 respectively, if exactly five and four of his six selected numbers match the winning numbers. What is the expected value of the amount a player wins in one game?

Let x be the amount that a player wins in one game. To answer the question above, we need to find:

P(x=1,200,000)

P(x= 800)

P(x=35)

P(x=0)

We’ll find these probabilities using combinations.

P(x=1,200,000) = 1/49C6 = .000000072

P(x=800) = (6C5)*(43C1)/49C6 = .000018

P(x=35) = (6C4)*(43C2)/ 49C6 = .00097

P(x=0)= 1- .000000072 - .000018 - .00097 = .999011928

We can then put this into a probability distribution table:

x	1,200,000	800	35	0
P(x)	.000 000 072	.000 018	.000 97	.999011928

1.     What is the expected value of the winnings per game?

2.    Discussion: Go to your discussion board for project 4 and answer the following questions in at least a paragraph. If the tickets cost 50 cents each, what is the expected loss per game? If a person plays 100 games over a period of years, how much will he expect to lose in total? Please explain how you arrived at your conclusions. Respond constructively to at least one other student.

Answer 1

Let x be the amount that a player wins in one game. To answer the question above, we need to find:

P(x=1,200,000)

P(x= 800)

P(x=35)

P(x=0)

We’ll find these probabilities using combinations.

P(x=1,200,000) = 6C6 /49C6 = .000000072

P(x=800) = (6C5)*(43C1)/49C6 = .000018

P(x=35) = (6C4)*(43C2)/ 49C6 = .00097

P(x=0)= 1- .000000072 - .000018 - .00097 = .999011928

1.) Expected value of winning per game is the sumproduct of the probabilities and the winning amt

= 1,200,000*0.000000072 +800*0.000018+35*0.00097+0*0.999011928

= 0.13475

So the expected pay off is 0.13475$

2.) If the tkt cost is 0.50 $ the the expected loss= 0.5-expected payoff = 0.5-0.13475 =0.36525$

If a person plays 100 games then the expected loss will be = 100*0.36525 = 36.525$ (i..e. total games * loss per game as the games are mutually exclusive)

Hope the above answer has helped you in understanding the problem. Please upvote the ans if it has really helped you. Good Luck!!