General Comments Regarding Odds
The calculations of odds and probability use a mathematical function called the hypergeometric distribution. This distribution applies to cases where a set of objects 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 the Minnesota State Lottery games. Remember, that odds are the inverse of probability so if a formula has calculated the probability, you divide one by that number to get the odds.
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.
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 42 possible outcomes, making the probability 1/42.
On the white balls, when the first ball is drawn there is a 1 in 55 chance that it matches the first number on your ticket, a 1 in 55 chance that it matches the second number, and so on. All of these outcomes are equally good, so there is a 5 in 55 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 54 balls left in the machine, so you have a 4 in 54 chance of also matching the second ball drawn. Once you've done this, you have a 3 in 53 chance of matching the third ball, a 2 in 52 chance of also matching the fourth ball, and a 1 in 51 chance of matching the last ball drawn. The probability of matching all five balls, then is (5/55) x (4/54) x (3/53) x (2/52) x (1/51), which works out to .00000028746 or odds of 1 in 3,478,671.
To get the probability of winning the jackpot, multiply this by the probability of matching the power ball, and to get .0000000068442 or odds of 1 in 146,107,962.
2) Match 5 without the power ball.
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 3,478,671. 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 41 in 42, so the odds of winning the $100,000 prize are 1/3,478,671 x 41/42, or 1 in 3,563,609.
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 42. In actuality, it is the odds of matching the red ball multiplied by the odds of not matching any of the white balls.
Hot Lotto Odds
Odds for Hot Lotto are figured exactly the same way as the odds for Powerball, except, of course, that the number of balls is different.
As with Powerball, 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, 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 and Powerball.
Northstar Cash Odds
Northstar Cash odds are calculated using the same general principles as Gopher 5, Powerball, 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.
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