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Counting objects satisfying certain criteria.
Number of ways to order n items.
| n! = n * (n-1) * (n-2) * ... * 2 * 1 for integer n>0 and 0! = 1 |
Card Example: Number of ways to shuffle a deck of cards.
| 52! = 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000 |
Number of ways r samples can be chosen from a set of size n, aka n choose r, is
| nCr(n, r) = n! / (r! * (n-r)!) |
Card Example: Number of ways to deal a hand of 7 cards.
| nCr(52, 7) = 52! / (7! * 45!) = 133,784,560 |
The number of ways Σiri samples can be partitioned into subsets of type i with size ri.
| Cr(ri) = (Σiri)! / Πi(ri!) |
Card Example: Number of ways to deal 4 people each a hand of 7 cards.
| Cr(7, 7, 7, 7, 24) = (7 + 7 + 7 + 7 + 24)! / (7! * 7! * 7! * 7! * 24!) = 201,474,727,133,525,966,905,424,640,000 |
| If the player ordering not important, then number of unique deals is given by dividing the above by 4!. |
The hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of draws from a finite population without replacement. I give my own take on this type of lottery below; by trade I'm not a statistician so it might be a good idea to have a look here too.
A population of size N contains R successes. A sample of size n is taken without replacement. It is desired that the sample contains exactly r successes.
| Number of desirable samples: | nCr(R, r) * nCr(N-R, n-r) |
| Probability a sample is desirable: | nCr(R, r) * nCr(N-R, n-r) / nCr(N, n) |
| ie |
Card Example: The odds of dealing exactly 2 aces in a hand of 7 cards.
| Deck of 52 cards contain 4 aces Hand of 7 cards with 2 aces |
N=52, R=4 n=7, r=2 |
|
| The number of hands Probability of dealing |
nCr(4, 2) * nCr(48, 5) = 10,273,824
nCr(4, 2) * nCr(48, 5) / nCr(52, 7) = 594 / 7,735 |
|
| The odds of dealing | ever so slightly worse than 1 in 13 | |
Lottery Example: There are a total of 54 unique balls. The player selects 6 balls on their ticket. The draw machine selects randomly 6 balls. Winning outcomes are: match 6, match 5, match 4, or match 3.
| The number of balls in the game
The number of balls the player chooses The number of balls drawn The number of balls to match |
N = 54
R = 6 n = 6 r = 6, 5, 4, 3 | ||||
| Winning outcome | match exactly 6 | match exactly 5 | match exactly 4 | match exactly 3 | |
| The odds | 1 in 25,827,165 | 1 in 89,678 | 1 in 1,527 | 1 in 75 | |
The Lotto Texas has the above configuration. Lotteries can sometimes employ several independent machines. Here combinations and probabilities are simply multiplied out, and I place the contingency tables side by side. An example would be EuroMillions.
A population of size ΣiRi exhibits a partitioning into subsets of type i with size Ri. A sample of size Σiri is taken without replacement. It is desired that the sample exhibits a partitioning into subsets of type i with size ri.
| Number of desirable samples: | Πi nCr(Ri, ri) |
| Probability a sample is desirable: | Πi nCr(Ri, ri) / nCr(ΣiRi, Σiri) |
Card Example: The odds of dealing exactly 1 ace and 3 picture cards in a hand of 7 cards.
| The deck contains 4 aces, 12 picture cards and 36 other cards
The hand contains 1 ace, 3 picture cards and 3 other cards |
R1 = 4, R2 = 12, R3 = 36
r1 = 1, r2 = 3, r3 = 3 |
|
| The number of hands
Probability of dealing |
nCr(4, 1) * nCr(12, 3) * nCr(36, 3) = 6,283,200
nCr(4, 1) * nCr(12, 3) * nCr(36, 3) / nCr(52, 7) = 660 / 14,053 |
|
| The odds of dealing | slightly worse than 1 in 21 | |
Lottery Example: There are a total of 49 unique balls. The player selects 6 balls on their ticket. The draw machine selects randomly 6 main balls, and then 1 bonus ball. Winning outcomes are: match 6, match 5 and bonus, match 5, match 4 or match 3. Note that match 6, match 4 and match 3 don't involve the bonus ball. Match 5 is really match 5 and not the bonus.
| Winning outcome | match exactly 6 main | match exactly 5 main and 1 bonus | match exactly 5 main and 0 bonus | match exactly 4 main | match exactly 3 main | |
| The odds | 1 in 13,983,816 | 1 in 2,330,636 | 1 in 55,491 | 1 in 1,032 | 1 in 57 | |
| The column sums give the ball type totals (main, bonus, other): this is either (6,0,43) or (6,1,42) depending if the bonus ball is in play. The row sums give the ticket choice totals (selected, rejected): always (6, 43). Finally, the sum of all elements is the number of balls in play: always 49. | ||||||
| Consequently, the number of possible draws is Cr(6,0,43) = 13,983,816 without the bonus ball, or Cr(6,1,42) = 601,304,088 with the bonus ball. The number of possible tickets is Cr(6,43) = 13,983,816. | ||||||
The UK Lotto has the above configuration.