I remember when New York State began it’s Lotto game. I was in eighth grade. In Lotto you would pick six numbers out of 40. On Saturday night there would be a televised drawing. Six numbers plus a supplemental number would be drawn and you would win various prizes based on how many numbers you matched. On the back of the betting slips you could see the odds of winning each prize: very large and precise numbers.
I liked math but at that age I didn’t even know what probability was. I really wanted to know how those odds were calculated and that was the beginning of my interest in probability. There are a couple of ways to calculate the odds and some people like to use formulas but I think that a better way for someone who really wants to understand the logic behind these calculations is to use the simple principles you learn very early in any probability course: the multiplication principle and the addition principle What makes it hard to derive these odds that these principles need to be applied to a decision tree that become somewhat complex. When dealing with any complex situation organization is the key to success. In this slides ahead we’ll use a decision tree.
The decision occurs each time a number is drawn. Either it matches a number you selected or it doesn’t. In Powerball there are five numbers drawn without replacement from a total of 69.
Then there is a red Powerball drawn from a separate set of 26. As mentioned the basic principles of probability that apply are very simple. Keeping track is complicated. This is essentially a bookkeeping problem so we need to be organized.
For every point on the decision tree we’ll need to keep careful track of the odds going into and out of it. I’ll use the following structure to help. On the top balls matched vs. balls drawn so far. It’s basically for keeping score.
In the center the odds of getting to this point, and at the bottom a tally of the balls remaining from which the odds pertaining to the next draw follow. Essentially this structure describes the past present and future as stages of the drawing progress. At the top is the present: at any stage of the game we’ve drawn a certain number of balls and we’ve matched a certain number of them. This is a scorecard. In the center is the past. The odds of getting to this point are determined by adding up the odds from the previous step or steps that got us here.
At the bottom is the future. The odds of matching or missing on the next draw are determined by figuring out how many balls remaining are potential matches or misses. We’ll begin by making a box that describes the situation just before the game starts.
We haven’t drawn or matched anything yet. As for the odds, every game starts with 100 percent probability at the beginning. As for the next draw, we have 69 balls to be drawn with five potential matches and 64 potential misses in that set, and we’ll ignore the Powerball until the last step. Now we draw the first ball. This is the easy step. It’s not hard to understand that with five potential matches and 64 potential misses in the draw we have 5 out of 69 chance of matching and 64 out of 69 chance of missing.
So we generate two boxes for the next step. Follow the arrows. The hundred percent chance is multiplied by either of the two possibilities for the next step that’s the multiplication principle. Then arrows are drawn to the next step. Fairly straightforward so far but things are about to get intricate and we’ll try to manage that in an organized way.
The next step gets a little more complicated. For every arrow we multiply the previous odds by the match or miss odds and then add all the incoming arrows. That’s the multiplication principle followed by the addition principle.This guarantees that all the probabilities sum to 1, which is expected because they cover all the possibilities for the number of balls matched at that stage from zero matches on up to the number of balls drawn so far. So now on to the fourth ball. We have four boxes leading into five.
In each case we take the odds for the current step at the center, multiply by the odds at the bottom for a match vs no match, draw arrows to the appropriate next step based on how many balls will be matched in that case and add the incoming arrows. Some patterns are emerging and these patterns lead to formulas. Without discussing in too much detail, you can see that there are products of steadily decreasing quantities like 5 times 4 times 3 or 69 times 68 times 67 and these formulas involve factorials. Also many of the odds have an additional factor in front of them like four or six and those factors come from Pascal’s Triangle. Now drawing the fifth ball, I have to decrease the size of the boxes still more and then finally the Powerball.
Lots of multiplication and division involved in these few steps, and I’ve also converted the fractional odds at the end to have a one in the numerator since we humans tend to interpret odds better when they are expressed as one in some larger number like 100 million. In Powerball there are 12 possible outcomes and ten of these pay a cash prize. The two outcomes which don’t pay a cash prize account for a large majority of the outcomes. They occur when you match a single white ball at 1 in 3.68, and when you match nothing at all one in 1.53. To find the odds of winning any prize at all, add the odds for these two conditions: 1/1.53 + 1/3.68.
Subtract that total from one and it comes out to one in 13.18 . The cash prizes are usually many times smaller than the actual odds would imply is a fair payout. To mitigate this there is a prize multiplier that can multiply your winnings but you have to pay extra for that. The exception is for the jackpot which can become substantially larger than fair odds would imply since it keeps increasing until there is a winner and if you can be sure you won’t share the prize with another winner it’s actually a good bet when that situation occurs except that that good bet pays off only for a tiny handful of lucky people. I remember some organized efforts to buy up every combination in lotteries when the payouts exceeded the cost of all possible tickets but the logistics involved along with the risk of sharing the prize with someone else made this prohibitive.
And once again thanks for watching. I hope this video took some of the mystery out of that odds table.