For a multistate lottery, the following probability distribution represents the

Question

For a multistate lottery, the following probability distribution represents the cash prizes of the lottery with their corresponding probabilities. Complete parts (a) through (c) below. x (cash prize, $) P(x) Grand prize 0.00000000617 200,000 0.00000037 10,000 0.000001745 100 0.000138611 7 0.004471829 4 0.008147354 3 0.01353382 0 0.97370626483 (a) If the grand prize is $13 comma 000 comma 000 , find and interpret the expected cash prize. If a ticket costs $1, what is your expected profit from one ticket? To find the expected cash prize, recall that the expected value of a discrete random variable is the mean of the discrete random variable. The mean of a discrete random variable is given by the formula below. mu Subscript Upper X equalsSummation from nothing to nothing left bracket x times Upper P left parenthesis x right parenthesis right bracket Calculate mu Subscript Upper X using this definition, rounding to the nearest cent. mu Subscript Upper X equals Summation from nothing to nothing left bracket x times Upper P left parenthesis x right parenthesis right bracket mu Subscript Upper X equals $13 comma 000 comma 000 (0.00000000617)plus$ 200 comma 000 left parenthesis 0.00000037 right parenthesis plus $10,000(0.000001745)plus$100(0.000138611)plus$7(0.004471829 ) plus $ 4 left parenthesis 0.008147354 right parenthesisplus$3(0.01353382)plus$0(0.97370626483 ) mu Subscript Upper X equals $0.29 Thus, the expected cash prize is $0.29 . This means that, on average, you will win $ 0.29 per lottery ticket. To find your expected profit, subtract the cost of the ticket from the expected cash prize. $0.29 minus$1equalsminus$0.71 Therefore, your expected profit from one ticket is minus $0.71 . (b) To the nearest million, how much should the grand prize be so that you can expect a profit? Assume nobody else wins so that you do not have to share the grand prize. For simplicity, first find the grand prize value that will lead to an expected profit of zero. Then we'll round up to the nearest million to find the grand prize value that will give an expected profit. If the expected profit is zero, this means that mu Subscript Upper X minus1equals 0. Use this equation to find the grand prize. Now solve the equation. First write out mu Subscript Upper X minus1 using the equation mu Subscript Upper X Baseline equals Summation from nothing to nothing left bracket x times Upper P left parenthesis x right parenthesis right bracket . Then simplify. mu Subscript Upper X minus 1 equals $G(0.00000000617 )plus$ 200 comma 000 left parenthesis 0.00000037 right parenthesisplus$ 10 comma 000 left parenthesis 0.000001745 right parenthesis plus $100(0.000138611)plus$7(0.004471829)plus$ 4 left parenthesis 0.008147354 right parenthesis plus $ 3 left parenthesis 0.01353382 right parenthesisplus$0(0.97370626483)minus 1 mu Subscript Upper X minus 1 equals $G(0.00000000617 )plus$(negative 0.790195221 ) Therefore, the equation is $G(0.00000000617 )minus$0.790195221equals0. Solve for G by adding 0.790195221 to both sides and then dividing both sides by 0.00000000617 . Round to the nearest dollar. $G(0.00000000617 ) equals 0.790195221 G equals $128 comma 070 comma 538 Therefore, if a grand prize of approximately $128 comma 070 comma 538 gives an expected profit of zero, to the nearest million, a grand prize of $129 comma 000 comma 000 has an expected profit greater than zero. (c) Does the size of the grand prize affect your chance of winning? Explain. To answer this question, look carefully at the given chart above. Question is complete.

Explanation / Answer

(a) The grand prize = $ 13,000,000. With this value, the probability distribution table of the prize money is as below:

The third column shows the values of x*P(x)

The expected cash prize = x*P(x) = sum of the values in the 3rd column = $ 0.29001478 ~ $ 0.29

If a ticket costs $1, the expected profit = 0.29 - 1 = - $ 0.71

(b) Let the grand prize be donoted by G. To expect a profit, we should have x*P(x) -1 > 0

We can solve this equation as x*P(x) -1 = 0

G * (0.00000000617) + 200,000 * (0.00000037) + 10,000 * (0.000001745) + 100* (0.000138611) + 7 * (0.004471829) + 4 * (0.008147354) + 3 * (0.01353382) + 0 * (0.97370626483) - 1 = 0

So, G * (0.00000000617) = 0.790195221

G = $ 128,070,538

The nearest million value is 129

So a grand prize money of $ 129,000,000 will ensure a positive expected profit.

(c) The Grand prize doesn't affect the chances of winning, because the probability distribution is fixed, but it affects the chances of winning a profit, because a large grand prize, even with a small probability of winning the grand prize, will affect the expected winnings from the lottery. A large grand prize will lead to a higher expected profit from the lottery and vice versa.

x P(x) x*P(x)          13,000,000 0.00000000617         0.08021000                200,000 0.00000037         0.07400000                  10,000 0.000001745         0.01745000 100 0.000138611         0.01386110 7 0.004471829         0.03130280 4 0.008147354         0.03258942 3 0.01353382         0.04060146 0 0.973706265                            -  

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.