Mega Million Odds
Background:
Mega Millions is a lottery where you choose 6 numbers. The first five numbers must be chosen from the
numbers 1 to 75 and the last number (called the MegaBall Number) must be chosen from a different set
of numbers 1 to 15.
Below is the list of ways that you can win a prize and the prize amount
Choose 5
Numbers from
1 to 75
Match all 5
Match all 5
Match 4 out of 5
Match 4 out of 5
Match 3 out of 5
Match 3 out of 5
Match 2 out of 5
Match 1 out of 5
No Match
Choose 1
Number from
1 to 15
Match
No Match
Match
No Match
Match
No Match
Match
Match
Match
Prize
Jackpot
Win $1,000,000
Win $5,000
Win $500
Win $50
Win $5
Win $5
Win $2
Win $1
Calculating the Odds:
In order to calculate the odds of winning any prize, we first need to find out the number of possible ways
to choose the 5 numbers from 1 to 75 and the 1 number from 1 to 15.
Total number of possible combinations = choosing 5 numbers out of 75 x choosing 1 number out of 15
75
Total number of possible combinations = 5𝐶
x
15
1𝐶 = 17,259,390 x 15 = 258,890,850
Odds of Winning the Jackpot:
In order to win the Jackpot, you will need to match all of 5 numbers as well as the megaball number
5
The number of ways to match the 5 winning numbers is 5𝐶 and the number of ways to match the
1
winning megaball number is 1𝐶 . So total number of ways to match the winning combination of the 5
5
1
numbers and the megaball = 5𝐶 x 1𝐶 = 1.
So the odds of winning the jackpot = 1 out of 258,890,850
Probability of winning the jackpot = 0.00000000386
Odds of Matching the Five Numbers:
5
The number of ways to match the 5 winning numbers is 5C. Since there is no match with the megaball
14
number, it must be one of the other 14 numbers that are not chosen. This can happened in 1C ways. So
5
14
the number of ways to match just the five numbers = 5𝐶 x 1𝐶 = 1 x 14 = 14
So the odds of Matching just the five numbers = 14 out of 258,890,850 or 1 out of 18,292,204
Probability of Matching just the five numbers = 0.00000005408
Odds of Matching Four out of Five Numbers and the Megaball:
5
The number of ways to match the 4 out the 5 winning numbers is 4𝐶 = 5 ways. Now the other remaining
70
number should be 1 out of the 70 remaining numbers. This can be done in 1𝐶 = 70 ways. So the
5
70
number of ways to choose 4 out of the 5 numbers = 4𝐶 x 1𝐶 = 5 x 70 = 350.
1
The number of ways to match the winning megaball number is = 1𝐶
5
The number of ways to match 4 out 5 numbers and the megaball number = 4𝐶
x
70
1𝐶
x 11𝐶
= 350
The odds of matching 4 out 5 numbers and the megaball number = 350 out of 258,890,850 or 1 out of
739688
Probability of matching 4 out 5 numbers and the megaball number = 0.00000135192
Odds of Matching Four out of Five Numbers without the Megaball Ball:
5
The number of ways to match the 4 out the 5 winning numbers is 4𝐶 = 5 ways. Now the other remaining
70
number should be 1 out of the 70 remaining numbers. This can be done in 1𝐶 = 70 ways. So the
5
70
number of ways to choose 4 out of the 5 numbers = 4𝐶 x 1𝐶 = 5 x 70 = 350.
Since there is no match with the megaball number, it must be one of the other 14 numbers that are not
5
14
chosen. This can happened in 1𝐶 ways. So the number of ways to match just the five numbers = 5𝐶 x
14
1𝐶 = 1 x 14 = 14
5
5
70
14
The number of ways to match 4 out 5 numbers without the Megaball = 4𝐶 x 1𝐶 x 5𝐶 x 1𝐶 = 350
x 14 = 4,900
The odds of matching 4 out 5 numbers without the Megaball = 4900 out of 258,890,850 or 1 out of 52,835
Probability of matching 4 out 5 numbers without the Megaball = 0.00001892690
Odds of Matching Three out of Five Numbers with the Megaball Ball:
5
The number of ways to match the 3 out the 5 winning numbers is 3𝐶 = 10 ways. Now the 2 other
70
numbers should be 2 out of the 70 remaining numbers. This can be done in 2𝐶 = 2415 ways. So the
5
70
number of ways to choose 3 out of the 5 numbers = 3𝐶 x 2𝐶 = 10 x 2415 = 24,150 ways.
1
The number of ways to match the winning megaball number is = 1𝐶
5
The number of ways to match 3 out 5 numbers with the Megaball = 3𝐶
x
70
2𝐶
x 11𝐶
= 24,150
The odds of matching 3 out 5 numbers with the Megaball = 24150 out of 258,890,850 or 1 out of 10,720
Probability of matching 3 out 5 numbers with the Megaball = 0.00009328256
Odds of Matching Three out of Five Numbers without the Megaball Ball:
5
The number of ways to match the 3 out the 5 winning numbers is 3𝐶 = 10 ways. Now the 2 other
70
numbers should be 2 out of the 70 remaining numbers. This can be done in 2𝐶 = 2415 ways. So the
5
70
number of ways to choose 3 out of the 5 numbers = 3𝐶 x 2𝐶 = 10 x 2415 = 24,150 ways.
Since there is no match with the megaball number, it must be one of the other 14 numbers that are not
14
chosen. This can happened in 1𝐶 ways.
5
The number of ways to match 3 out 5 numbers without the Megaball = 3𝐶
x
70
2𝐶
x
14
1𝐶 = 338100
The odds of matching 3 out 5 numbers without the Megaball = 338100 out of 258,890,850 or 1 out of 766
Probability of matching 3 out 5 numbers without the Megaball = 0.00130595577
Odds of Matching Two out of Five Numbers with the Megaball Ball:
5
The number of ways to match the 2 out the 5 winning numbers is 2𝐶 = 10 ways. Now the 3 other
70
numbers should be 3 out of the 70 remaining numbers. This can be done in 3𝐶 = 54,740 ways. So the
5
70
number of ways to choose 2 out of the 5 numbers = 2𝐶 x 3𝐶 = 10 x 54,740 = 547,400 ways.
1
The number of ways to match the winning megaball number is = 1𝐶
5
The number of ways to match 2 out 5 numbers with the Megaball = 2𝐶
x
70
3𝐶
x 11𝐶
= 547,400
The odds of matching 2 out 5 numbers with the Megaball =547,400 out of 258,890,850 or 1 out of 473
Probability of matching 2 out 5 numbers with the Megaball = 0.00211440458
Odds of Matching One out of Five Numbers with the Megaball Ball:
5
The number of ways to match the 1 out the 5 winning numbers is 1𝐶 = 5 ways. Now the 4 other numbers
70
should be 4 out of the 70 remaining numbers. This can be done in 4𝐶 = 916,895 ways. So the number
5
70
of ways to choose 1 out of the 5 numbers = 1𝐶 x 4𝐶 = 5 x 916,895 = 4,584,475 ways.
1
The number of ways to match the winning megaball number is = 1𝐶
5
The number of ways to match 1 out 5 numbers with the Megaball = 1𝐶
x
70
4𝐶
x 11𝐶
= 4,584,475
The odds of matching 1 out 5 numbers with the Megaball =4,458,475 out of 258,890,850 or 1 out of 56
Probability of matching 1 out 5 numbers with the Megaball = 0.017708138
Odds of Matching Zero out of Five Numbers with the Megaball Ball:
5
The number of ways to match the 0 out the 5 winning numbers is 0𝐶 = 1 ways. Now all the 5 numbers
70
should be from the 70 remaining numbers. This can be done in 5𝐶 = 12,103,014 ways. So the number
70
5
of ways to choose 0 out of the 5 numbers = 0𝐶 x 5𝐶 = 1 x 12,103,014 = 12,103,014 ways.
1
The number of ways to match the winning megaball number is = 1𝐶
5
The number of ways to match 0 out 5 numbers with the Megaball = 0𝐶
x
70
5𝐶
x 11𝐶
= 12,103,014
The odds of matching 0 out 5 numbers with the Megaball =12,103,014 out of 258,890,850 or 1 out of 21
Probability of matching 0 out 5 numbers with the Megaball = 0.04674948535
Odds of winning Something:
The table below provides all of the odds and probabilities for winning each of the nine prizes. Adding all of
the nine probabilities gives the probability of winning something = 0.0679916
Odds of winning something = 1 out of 14.71
Probability of not winning anything = 1- Probability of winning something = 0.932008
Odds of winning nothing at all = 1 out of 1.07295
As the table shows, there is a 93.20% chance that you will not win anything.