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.