How to calculate lottery odds yourself

Lottery odds are not a secret and not an estimate. They are a counting exercise you can do on paper, and every figure a lottery publishes can be reproduced from the ball matrix alone. Here is the whole method, with the arithmetic shown.

Step 1: order does not matter

A ticket matches whether or not the balls arrive in your order, so the count you need is combinations, not permutations. The number of ways to choose k items from n is

C(n, k) = n! / [ k! × (n−k)! ]

For Mega Millions' five white balls from 70, the fraction is easier written out and cancelled:

C(70, 5) = (70 × 69 × 68 × 67 × 66) / (5 × 4 × 3 × 2 × 1) = 12,103,014

Dividing by 5! is what removes the orderings: every set of five numbers can be arranged 120 ways, and all 120 win the same prize.

Step 2: multiply by the second pool

The bonus ball comes from a separate machine, so it is an independent choice and the counts multiply. This is the step people get wrong most often — the bonus ball is not one of the five, and it can duplicate one of them.

Step 3: the other prize tiers

For a partial match you need to count two things at once: how many of your five numbers hit, and how many missed. If m of your five match, then m came from the 5 drawn balls and (5−m) came from the (N−5) balls that were not drawn:

P(exactly m white) = C(5, m) × C(N−5, 5−m) / C(N, 5)

Then multiply by 1/B if you also need the bonus ball, or (B−1)/B if you must miss it.

Worked example: 4 white balls plus the Powerball

C(5, 4) × C(64, 1) / C(69, 5) × 1/26 = 5 × 64 / 11,238,513 × 1/26 = 1 in 913,129

Powerball publishes 1 in 913,129.18 for that tier. The formula reproduces it exactly, and the same two lines of arithmetic generate every other row of the official chart.

Step 4: the overall odds

"Odds of winning any prize" is just the sum of the winning tiers' probabilities, inverted. Adding the nine rows above gives 1 in 24.87 for Powerball; the same sum for Mega Millions gives 1 in 23.07. Both match the official statements ("about 1 in 25" and "1 in 23") because they are the same calculation.

Four mistakes to avoid

• Adding when you should multiply. Independent stages multiply. The chance of matching five white balls and the bonus is the product of the two, not the sum.
• Using permutations. Forgetting to divide by 5! inflates the count by 120× and gives an "odds" figure in the tens of billions.
• Treating "1 in 292 million" as a percentage. It is 0.000000342%. Writing it as a percentage is a good way to feel the size of it.
• Assuming odds accumulate. Buying 10 tickets makes your chance 10 in 292,201,338, not 1 in 29,220,134 for each of ten independent shots at the same prize — and playing every week for fifty years buys around 2,600 draws against a 292 million denominator.

Reference: every matrix these two games have used

Both lotteries have changed their ball pools repeatedly, and every change moved the jackpot odds. This is also why frequency statistics must never be mixed across eras.

The trend is unmistakable: with two exceptions, every revision made the jackpot harder to win. The reasoning behind that is worth a page of its own.

Now do it for your own state game

The method is identical for any pick-5 or pick-6 game — swap N, k and B and turn the handle. If your result matches the odds on the back of the play slip, you have understood the game completely. If a website's numbers disagree with your arithmetic, trust the arithmetic.