Hypergeometric Distribution

Odds for specific scratch games and lotto games are listed with information about each individual game.

# Figuring the Odds

Odds for specific scratch games and lotto games are listed with information about each individual game.

## Scratch Games, Raffle, and Progressive Print-N-Play

These games feature a predetermined number of tickets and a predetermined number of winners. The odds are simply determined by calculating the ratio of winning tickets to total tickets. If, for example, there are five top prizes out of 1,000,000 tickets, the odds of winning one of these prizes is 5 divided by 1,000,000, or 1 in 200,000.

#### Lotto Games Odds

For  most lotto games, the calculation of odds and probability uses a mathematical function called the hypergeometric distribution. The exceptions are Daily 3 (see below) and games such as the Minnesota Millionaire Raffle and Progressive Print-N-Play that feature a fixed, predetermined number of tickets.

The hypergeometric distribution applies to cases where a set of object is divided into two sets. If we randomly draw r objects from this set, we want to know the probability that our draw will include exactly k objects of one of the two sets. A math textbook (such as "Feller, An Introduction to Probability Theory and its Application") or your local library can probably supply you with information about probability theory. In the descriptions below, these theories are applied to Minnesota State Lottery games. Remember that odds are the inverse of probability so if a formula has calculated the probability, you divide 1 by that number to get the odds.

#### Powerball Odds

1) Match 5 + Powerball

This is a two-part problem. The probability of matching all five white balls and the red ball is obtained by multiplying the probability of matching the five white balls by the probability of matching the red ball. These are unrelated, or independent, events. The outcome on the red ball has nothing to do with the white ball outcome, or vice versa.

The probability of matching the red ball is simple. There is one correct match and 35 possible outcomes, making the probability 1/35.

On the white balls, when the first ball is drawn there is a 1 in 59 chance that it matches the first number on your ticket, a 1 in 59 chance that it matches the second number, and so on. All of these outcomes are equally good, so there is a 5 in 59 chance that the first ball drawn matches a number on your ticket. If your ticket includes the first number you need to find the probability of matching the second number. You have four numbers left on your ticket, and there are 58 balls left in the machine, so you have a 4 in 58 chance of also matching the second ball drawn. Once you've done this, you have a 3 in 57 chance of matching the third ball, a 2 in 56 chance of also matching the fourth ball, and a 1 in 55 chance of matching the last ball drawn. The probability of matching all five balls, then is (5/59) x (4/58) x (3/57) x (2/56) x (1/55), which works out to .00000019974 or odds of 1 in 5,006,386.

To get the probability of winning the jackpot, multiply this by the probability of matching the Powerball, to get .000000005707 or odds of 1 in 175,223,510.

2) Match 5 without the Powerball.

This is one of the most common questions we get. People assume that it should simply be the odds of matching the five white balls, or 1 in 5,006,386. What they forget is that to win this prize you must also not match the red ball. Your chances of not matching the red ball are 34 in 35, so the odds of winning the \$1,000,000 prize are 1/5,006,386 x 34/35, or 1 in 5,153,633.

3) Match 0 and Powerball (\$3 prize)

Here people assume that it should be just the odds of matching the red ball, or 1 in 35. In actuality, it is the odds of matching the red ball multiplied by the odds of not matching any of the white balls.

#### Mega Millions Odds

Mega Millions odds are calculated in exactly the same way as Powerball odds with formulas changed to account for the different number of balls—75 possibilities (rather than Powerball’s 59) for the white balls and 15 possibilities for the gold Mega Ball instead of Powerball’s 39. The probability of matching the Mega Ball is 1 in 15.  Matching all five of the white balls would be calculated as (5/75) x (4/74) x (3/73) x (2/72) x (1/71), or .0000000579, which translates into odds of 1 in 17,259,390. The probability of winning the jackpot, therefore, is (1/17,259,390) x (1/15), or 0.00000000386, or 1 in 258,890,850.

#### Hot Lotto Odds

Odds for Hot Lotto are figured exactly the same way as the odds for Powerball and Mega Millions, except, of course, that the number of balls is different.

As with Powerball and Mega Millions, you must consider the probabilities of matching the first five numbers and of matching the Hot Ball separately. There are 19 possible outcomes for the Hot Ball, making the probability of matching it 1 in 19. Similarly, when the first white ball is drawn, there is a 1 in 39 chance it matches the first number on your ticket, a 1 in 39 chance that it matches the second number on your ticket, and so on, producing a 5 in 39 chance that the first ball drawn matches a number on your ticket. Again, using the same method as Powerball and Mega Millions, there's a 4 in 38 chance that the second ball drawn matches a number on your ticket, a 3 in 37 chance that the third ball drawn is a match, and so on. The probability of matching all five balls is (5/39) x (4/38) x (3/37) x (2/36) x (1/35), or 0.000001736, or 1 in 575,757. To obtain the probability of winning the jackpot, we multiply by the probability of matching the Hot Ball, resulting in a probability of 0.000000091, or 1 in 10,939,383.

To find the odds of winning the \$10,000 prize, multiply the probability of matching the five white balls (0.000001736) by the probability of not matching the Hot Ball (18 in 19), getting 0.000001645, or 1 in 607,744.

Odds of winning smaller prizes can be calculated using the same methods as we used for Gopher 5, Powerball and Mega Millions.

#### Gopher 5 Odds

When the first ball is drawn there is a 1 in 47 chance that it matches the first number on your ticket, a 1 in 47 chance that it matches the second number, and so on. All of these outcomes are equally good, so there is a 5 in 47 chance that the first ball drawn matches a number on your ticket. If your ticket includes the first number you need to find the probability of matching the second number. You have four numbers left on your ticket, and there are 46 balls left in the machine, so you have a 4 in 46 chance of also matching the second ball drawn. Once you've done this, you have a 3 in 45 chance of matching the third ball, a 2 in 44 chance of also matching the fourth ball, and a 1 in 43 chance of matching the last ball drawn. The probability of matching all five balls, then is (5/47) * (4/46) * (3/45) * (2/44) * (1/43). That calculation will give you a probability of .000000651916 or odds of 1 in 1,533,939. As stated above, odds are calculated by dividing 1 by the probability.

Calculating your chances of matching four of the five balls is more complicated since you need to take into account the various ways of not matching the fifth number. Here we take the 1,533,939 possible Gopher 5 outcomes and divide it by 5 (the number of ways of matching 4 out of 5) and again by 43 (the number of ways of not matching the fifth ball) to get odds of 1 in 7,304.

#### Northstar Cash Odds

Northstar Cash odds are calculated using the same general principles as Gopher 5, Powerball, Mega Millions and Hot Lotto.

With five numbers from 1 to 31 to select from, there is a 5 in 31 chance that the first number selected matches a number on your ticket. With four numbers left on your ticket, and 30 left to choose from, there is a 4 in 30 chance of matching the second number on your ticket. If you match the first two numbers, there’s a 3 in 29 chance of matching the third number, a 2 in 28 chance of matching the fourth number, and a 1 in 27 chance of matching the fifth number. The probability of matching all five numbers becomes (5/31) x (4/30) x (3/29) x (2/28) x (1/27), or 0.00000588543. The odds are 1/0.00000588543, or 1 in 169,911.

To calculate the odds of winning the smaller prizes, you need to take into account the number of combinations of numbers that don’t match the ones on your ticket. For example, to win the \$50 prize you must match four of the five numbers on your ticket. If the numbers on your ticket are 1, 2, 3, 4, and 5, you can win \$50 with five different combinations of these numbers: (1,2,3,4), (1,3,4,5), (1,2,4,5), (1,2,3,5), and (2,3,4,5). However, for each of these ways, there 26 possible non-matching fifth numbers — (1,2,3,4,6), (1,2,3,4,7), etc. — making for a total 130 possible drawing outcomes that will match four of the five numbers on your ticket. As we calculated above that there are 169,911 possible Northstar Cash outcomes, your probability of winning \$50 is 130/169,911, or 1 in 1,307.

The odds of winning the remaining prizes can also be calculated by counting the number of possible winning combinations and dividing by the total number of possible drawing outcomes (169,911). This counting process, however, can quickly become laborious, and use of the hypergeometric distribution becomes a much simpler alternative.

#### Daily 3 Odds

Daily 3 differs from the above games in that you’re selecting one number each from three different sets of numbers as opposed to selecting multiple numbers from a single set. It’s also different in that for some bets (like straight bets), the order in which the numbers are selected matters. The key to calculating Daily 3 odds is recognizing that the three numbers selected are independent—matching the first selection doesn’t change the odds of matching the second or third. The relevant mathematical function is called the binomial distribution, but you can calculate Daily 3 odds without resorting to mathematical formulas.

There are 10 possible ways the first number can be selected, 10 ways the second can be selected, and 10 ways the third number can be selected. The total number of possible outcomes, therefore, is 10 x 10 x 10, or 1000.  (If you’re not convinced of this, think of a smaller number of numbers, say, for example, there were only two numbers in each set rather than 10. Then list out all the possibilities: 1,1,1, or 1,1,2, or 1,2,1, or 2,1,1 or 2,2,1 or 2,1,2 or 1,2,2 or 2,2,2. There are eight possible combinations, which is 2 x 2 x 2.)

Now for a straight bet, there’s only one combination out of those thousand that matches your ticket, so the odds of winning a straight bet are 1 in 1000. But for box bets, there are multiple ways to win. Let’s say you’ve placed a 6-way box bet using the numbers 1,2, and 3. The following six combinations are all winners: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). Your odds of winning a 6-way box bet, therefore, are 6 in 1000, or 1 in 166.7.  For a 3-way box bet (where you repeat a number like 1,1,2), there would be three winning combinations: (1,1,2), (1,2,1) and (2,1,1), making the odds 3 in 1000 or 1 in 333.