Question

One state lottery has 1,000 prizes of $1; 130 prizes of $10; 20 prizes of $55; 5 prizes of $300; 2 prizes of $1,010; and 1 prize of $2,500. Assume that 31,000 lottery tickets are issued and sold $1

What is the lottery's expected profit per ticket?

What is the lottery's standard deviation of profit per ticket?

Answer 1

1,000 prizes of $1;

130 prizes of $10;

20 prizes of $55;

5 prizes of $300;

2 prizes of $1,010;

1 prize of $2,500

1,000 prizes of $1; 130 prizes of $10; 20 prizes of $55; 5 prizes of $300; 2 prizes of $1,010; and 1 prize of $2,500

No. of prizes f(i)	Amount in $ (xi)
1000	1
130	10
20	55
5	300
2	1010
1	2500

Assume that 31,000 lottery tickets are issued and sold $1

Thus , The lottery makes $31,000 on sales.

Its pay out is given by - No . of prizes * Amount of prize

No. of ticket f(i)	Prize in $ (xi)	xi*fi
1000	1	1000
130	10	1300
20	55	1100
5	300	1500
2	1010	2020
1	2500	2500

It pays out = 1000+1300+1100+1500+2020+2500 = 9420

It pays out = 9420

Total 31,000 lottery tickets are issued and sold $1

Ticket sold without giving any prizes ( i.e prize =0 ) = 31,000 - (1000+130 + 20 + 5 + 2 + 1)

                                                            = 29842

Prizes in $ (xi)	No. of ticket f(i)
0	29842
1	1000
10	130
55	20
300	5
1010	2
2500	1

We find Probability of each Prize value tickets P(x) = No. of ticket f(i) / 31,000

Let us define a variable X = Prize of ticket - selling price of each ticket ($1 )

                                             = xi - 1

Prizes in $ (xi)	No. of ticket f(i)	X=winnig	P(x)	P(x)*X
0	29842	-1	0.962645	-0.96265
1	1000	0	0.032258	0
10	130	9	0.004194	0.037742
55	20	54	0.000645	0.034839
300	5	299	0.000161	0.048226
1010	2	1009	6.45E-05	0.065097
2500	1	2499	3.23E-05	0.080613
Total	31000	3869	1	-0.69613

Thus lottery's expected profit per ticket is given by = - P(x)*X = - (-0.69613)

                          Lottery's expected profit per ticket = 0.69613

{ we have taken negative sign in (- P(x)*X ) because we defined variable X as Prize of ticket - 1 }

Total Profit = expected profit per ticket * Total Number of tickets sold = 0.69613 * 31000 = 21580

                   = $ 21,580

Now to find lottery's standard deviation of profit per ticket

Standard deviation =

Now = P(x)*(X - Mean(X) )²

Here Mean of X = E(X) = 0.69613

X=winnig	P(x)	P(x)*X	(X-E(X)) 2*P(x)
-1	0.962645	-0.96265	2.769391985
0	0.032258	0	0.015632129
9	0.004194	0.037742	0.289194154
54	0.000645	0.034839	1.832640149
299	0.000161	0.048226	14.32661702
1009	6.45E-05	0.065097	65.57564678
2499	3.23E-05	0.080613	201.6011679
3869	1	-0.69613	Total =286.4102901

Thus Var(X) =sum ( ( X- E(X) ) ²*P(x) )

                     =286.4102901

Hence , lottery's standard deviation of profit per ticket =

                                                                                       = 16.92366

And lottery's expected profit per ticket is 0.69613

And total profit is $ 21,580