Winning the Big One.... (2016 edition)
Last night's Powerball jackpot was $500 million, and there were no winners. Saturday's jackpot is over $700 mllion -- a new record.
The rules for both multi-state lotteries have changed since I posted this article four years ago. So let's update this a little, shall we?
Mega Millions: Players now pick 5 of 75 white balls, and one of 15 red balls. Drawings are still held on Tuesday and Friday nights. Cost to play is $1.
With the 75 white balls, what is the probability of getting all 5 of your numbers drawn? Let's figure it out. Draw the first ball from the pool of 75. Then draw the second ball from the remaining pool of 74. Then draw the third ball from the remaining pool of 73. Then draw the fourth ball from the remaining pool of 72. Then draw the fifth and final ball from the remaining pool of 71.
Mathematically, that would expressed thusly: 75 x 74 x 73 x 72 x 71 = 2,071,126,800. To 1 against.
Then divide this number by the number of permutations possible within those 5 numbers. If you're familiar with number theory and factorials, you know that there are 120 possibilities that the numbers can be drawn. (1 * 2 * 3 * 4 * 5 = 120.) Try it out: with the letters a, b, c, d, and e, how many five-letter combinations can you come up with? With one letter, there's one permutation -- a. With two letters, there are 2 permutations -- ab, ba. With 3 letters, there are 6 permutations -- abc, acb, bac, bca, cab, cba. With four letters, there are 24 permutations. And so on.
Back to the math: 2,071,126,800 / 120 = 17,259,390.
Now, bring in the draw for the red ball. A player has a 1-in-15 chance for their number to be drawn. So let's multiply the previous result by 15.
17,259,390 * 15 = 258,890,850.
There you have it. 258 million to one against.
Powerball: Players now pick 5 of 69 white balls, and one of 26 red balls. Drawings are still held on Wednesday and Saturday nights. Cost to play is $2.
For the white balls, the probability of getting all 5 of your numbers drawn is 69 x 68 x 67 x 66 x 65 = 1,348,621,560. To 1 against.
Divided by the number of permutations: 1,348,621,560 / 120 = 11,238,513
Multiplied by the 1-in-16 chances for the red ball: 11,238,512 * 26 = 292,201,338
Well there you go. 292 million to 1 against.
This can be expressed as a mathematical formula: ((w!/(w-d)!)/d!) * r, where w is the number of white balls, d for the number of white ball draws, and r for the number of red balls.
So, for Mega Millions, the formula is ((75!/70!)/5!) * 15, and for Powerball, the formula is ((69!/64!)/5!) * 26.
In short, the odds of you -- or anyone else -- winning is, in a nutshell, not bloody likely! This is why lotteries such as these are often called "a voluntary tax on those who are are bad at math". If you do play, do so responsibly. Don't spend more than you can comfortably afford to lose.
And, to paraphrase the announcer from "Hardware Wars": "You'll laugh, you'll cry, you'll kiss two bucks goodbye!"
The rules for both multi-state lotteries have changed since I posted this article four years ago. So let's update this a little, shall we?
Mega Millions: Players now pick 5 of 75 white balls, and one of 15 red balls. Drawings are still held on Tuesday and Friday nights. Cost to play is $1.
With the 75 white balls, what is the probability of getting all 5 of your numbers drawn? Let's figure it out. Draw the first ball from the pool of 75. Then draw the second ball from the remaining pool of 74. Then draw the third ball from the remaining pool of 73. Then draw the fourth ball from the remaining pool of 72. Then draw the fifth and final ball from the remaining pool of 71.
Mathematically, that would expressed thusly: 75 x 74 x 73 x 72 x 71 = 2,071,126,800. To 1 against.
Then divide this number by the number of permutations possible within those 5 numbers. If you're familiar with number theory and factorials, you know that there are 120 possibilities that the numbers can be drawn. (1 * 2 * 3 * 4 * 5 = 120.) Try it out: with the letters a, b, c, d, and e, how many five-letter combinations can you come up with? With one letter, there's one permutation -- a. With two letters, there are 2 permutations -- ab, ba. With 3 letters, there are 6 permutations -- abc, acb, bac, bca, cab, cba. With four letters, there are 24 permutations. And so on.
Back to the math: 2,071,126,800 / 120 = 17,259,390.
Now, bring in the draw for the red ball. A player has a 1-in-15 chance for their number to be drawn. So let's multiply the previous result by 15.
17,259,390 * 15 = 258,890,850.
There you have it. 258 million to one against.
Powerball: Players now pick 5 of 69 white balls, and one of 26 red balls. Drawings are still held on Wednesday and Saturday nights. Cost to play is $2.
For the white balls, the probability of getting all 5 of your numbers drawn is 69 x 68 x 67 x 66 x 65 = 1,348,621,560. To 1 against.
Divided by the number of permutations: 1,348,621,560 / 120 = 11,238,513
Multiplied by the 1-in-16 chances for the red ball: 11,238,512 * 26 = 292,201,338
Well there you go. 292 million to 1 against.
This can be expressed as a mathematical formula: ((w!/(w-d)!)/d!) * r, where w is the number of white balls, d for the number of white ball draws, and r for the number of red balls.
So, for Mega Millions, the formula is ((75!/70!)/5!) * 15, and for Powerball, the formula is ((69!/64!)/5!) * 26.
In short, the odds of you -- or anyone else -- winning is, in a nutshell, not bloody likely! This is why lotteries such as these are often called "a voluntary tax on those who are are bad at math". If you do play, do so responsibly. Don't spend more than you can comfortably afford to lose.
And, to paraphrase the announcer from "Hardware Wars": "You'll laugh, you'll cry, you'll kiss two bucks goodbye!"