Shall I play the Mega Millions Tonight? http://en.wikipedia.org/wiki/Mega_Millio

Question

Shall I play the Mega Millions Tonight?

http://en.wikipedia.org/wiki/Mega_Millions

The Friday ESTIMATED JACKPOT is $340 MILLION as of the last Draw Date in mid-July 2018. For a 2 dollar ticket these are the returns and their odds as listed below. For example, one would get your 2 dollars back with probability 1 in 56.

a) Substitute 340 millions for the jackpot and compute the average (expected, ? value). We assume here that there is only one possible winner. Remember that the probability of no prize ($0) is 1 minus the sum of probabilities listed in the table above.

b) Compute the standard deviation associated with this return (?). Based on your results, is it wise to invest the 2 dollars needed for the lottery tonight?

c) The largest registered jackpot is $656 millions, on March 30, 2012 (did you play?). Substitute this value in the table and repeat the calculation from part a (? and ?, although keep in mind that the probabilities have changed since then …, but keep the probabilities from this table so you don’t have to look up the old probabilities). Based on your results, would it be wise to invest 2 dollars when the jackpot becomes this large? Explain.

Explanation / Answer

Solution

Let x = prize money and p(x) = probability of getting the prize money x.

Back-up Theory

Expected Value, E(X) = ?[x.p(x)], summed over all possible values of x. ……………….(1)

Standard deviation, SD(X) = sqrtV(X) …………………………………………….....……….(2)

V(X) = E(X2) – {E(X)}2 …………………………………………………………………...........(3)

E(X2) = ?[x2.p(x)], summed over all possible values of x. ………………………………...(4)

Now, to work out the answer,

Preparatory Work

Prize ($) - x

Probability 1 in [p(x)]

x.p(x)

x^2.p(x)

3.40E+08

258890850

1.313294773

446520222.7

1.00E+06

18492204

0.054076842

54076.84233

5000

739688

0.006759607

33.79803377

500

52835

0.009463424

4.731711933

50

10720

0.004664179

0.233208955

5

766

0.006527415

0.032637076

5

473

0.010570825

0.052854123

2

56

0.035714286

0.071428571

1

21

0.047619048

0.047619048

0

0.930990541

0

0

Total

1

1.488690398

446574338.5

Part (a)

Expected Value = $1.49 ANSWER 1 [vide (1) under Back-up Theory and Col 3 of above table.]

Part (b)

V(X) = 446574336 [vide (3) and (4) under Back-up Theory and Col 3 and 4 of above table.]

Standard deviation = 21132 ANSWER 2 [vide (2) under Back-up Theory]

Since E(X) < 2, it is not wise to invest $2 on the ticket. ANSWER 3

Part (c)

With largest jackpot of 656 million incorporated,

Expected Value = $2.71 ANSWER 4

V(X) = 1662283577

Standard deviation = 40771 ANSWER 5

Since E(X) > 2, it is now wise to invest $2 on the ticket. ANSWER 6

Details of calculations

Prize ($) - x

Probability 1 in [p(x)]

x.p(x)

x^2.p(x)

6.56E+08

258890850

2.533886385

1662229469

1.00E+06

18492204

0.054076842

54076.84233

5000

739688

0.006759607

33.79803377

500

52835

0.009463424

4.731711933

50

10720

0.004664179

0.233208955

5

766

0.006527415

0.032637076

5

473

0.010570825

0.052854123

2

56

0.035714286

0.071428571

1

21

0.047619048

0.047619048

0

0.930990541

0

0

Total

1

2.70928201

1662283584

DONE

Prize ($) - x

Probability 1 in [p(x)]

x.p(x)

x^2.p(x)

3.40E+08

258890850

1.313294773

446520222.7

1.00E+06

18492204

0.054076842

54076.84233

5000

739688

0.006759607

33.79803377

500

52835

0.009463424

4.731711933

50

10720

0.004664179

0.233208955

5

766

0.006527415

0.032637076

5

473

0.010570825

0.052854123

2

56

0.035714286

0.071428571

1

21

0.047619048

0.047619048

0

0.930990541

0

0

Total

1

1.488690398

446574338.5

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