Probability of winning Mega Millions?? 5 white balls labeled (1-75) will be pick
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Question
Probability of winning Mega Millions??5 white balls labeled (1-75) will be picked randomly without replacement. and one "mega number" labeled (1-15) will be picked. each play cost $1
Winning Combination - Wins Get all 5 number and Mega - Jackpot Get 5 number without mega - 1,000,000$ get 4/5 number and mega - 5,000$ get 4/5 number without mega - $500 get 3/5 number with mega - $50 get 3/5 number without mega -$5 get 2/5 number with mega -$5 get 1/5 number with mega - $2 get 0/5 number and only mega $1
what's the probability of each winning? and how much does the jackpot have to be to make the game fair?
Probability of winning Mega Millions??
5 white balls labeled (1-75) will be picked randomly without replacement. and one "mega number" labeled (1-15) will be picked. each play cost $1
Winning Combination - Wins Get all 5 number and Mega - Jackpot Get 5 number without mega - 1,000,000$ get 4/5 number and mega - 5,000$ get 4/5 number without mega - $500 get 3/5 number with mega - $50 get 3/5 number without mega -$5 get 2/5 number with mega -$5 get 1/5 number with mega - $2 get 0/5 number and only mega $1
what's the probability of each winning? and how much does the jackpot have to be to make the game fair?
5 white balls labeled (1-75) will be picked randomly without replacement. and one "mega number" labeled (1-15) will be picked. each play cost $1
Winning Combination - Wins Get all 5 number and Mega - Jackpot Get 5 number without mega - 1,000,000$ get 4/5 number and mega - 5,000$ get 4/5 number without mega - $500 get 3/5 number with mega - $50 get 3/5 number without mega -$5 get 2/5 number with mega -$5 get 1/5 number with mega - $2 get 0/5 number and only mega $1
what's the probability of each winning? and how much does the jackpot have to be to make the game fair?
Explanation / Answer
Winnin Combination Wins Probability
5 White + Mega Jackpot 0.000000003863
5 White No Mega 1,000,000 0.00000005408
4 White + Mega 5,000 0.000001352
4 White No Mega 500 0.00001893
3 White + Mega 50 0.00009328
3 White No Mega 5 0.001306
2 White + Mega 5 0.002114
1 White + Mega 2 0.01771
0 White + Mega 1 0.04675
The number of ways 5 numbers can be randomly selected from a field of 75 is: 75C5 = 17,259,390
For each of these 17,259,390 combinations there are 15C1 = 15 different ways to pick the sixth number (the “Mega” number). The total number of ways to pick the 6 numbers is the product of these. Thus, the total number of equally likely Mega Millions combinations is 17,259,390 x 15 = 258,890,850
For jackpot, the number of ways the first 5 numbers on your lottery ticket can match the 5 White numbers is 5C5 = 1. The number of ways your final number can match the Mega number is: 1C1 = 1. The product of these is the number of ways you can win the Jackpot: 5C5 x 1C1 = 1. The probability of success is thus: 1/258,890,850 = 0.0000000038626316844.
The number of ways the 5 first numbers on your lottery ticket can match the 5 White numbers is 5C5 = 1. The number of ways your final number can match any of the 14 losing Mega numbers is: 14C1 = 14. (Pick any of the 14 losers.) Thus there are 5C5 x 14C1 = 14 possible combinations. The probability for winning $1,000,000 is thus 14/258,890,850 = .00000005407684358
The number of ways 4 of the 5 first numbers on your lottery ticket can match the 5 White numbers is 5C4 = 5. The number of ways your fifth initial number can match any of the 70 losing White numbers is 70C1 = 70. The number of ways your final number can match the Mega number is: 1C1 = 1. The product of these is the number of ways you can get this configuration: 5C4 x 70C1 x 1C1 = 350. The probability of success is thus: 350/258,890,850 = 0.00000135192109.
The remaining are done similarly and the probabilities are shown in the table above.
Total Probability for winning something = 0.06799361994
Probability of winning nothing = 1-0.06799361994 = 0.9320063801
To make game fair the expected values should equal.
$1*0.9320063801 = (1*0.04675)+(2*0.01771) + (5*(0.002114+0.001306)) + (50*0.00009328) + (500*0.00001893) + (5000*0.000001352) + (1000000*0.00000005408) + (x*0.000000003863)
0.9320063801 = 0.174239 + (x*0.000000003863), x = $196,160,336.552 for the game to be fair
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