The odds of a "Lotto" style lottery can be found with the formula: n! / (n - r)! r! where n is the highest numbered ball and r is the number of balls chosen. This is called in math a combination. An easier way to think about it is if there are 40 balls and 6 are chosen, there are 40 possible numbers that can come up first, leaving 39 that can come up second, then 38, 37, 36, and finally 35 on the final number. To find out how many numbers that is you multiply 40 ×39 ×38 ×37 ×36 × 35 = 2,763,633,600 making the odds 2 and a half billion to one.

Pretty slim odds, but luckily the order of the balls does not matter, so we can divide this number by how many ways these numbers can be arranged. There are six possibilities for the first ball, five for the second, 4 for the third, 3, 2, and one left over. That is 6 × 5 × 4 × 3 × 2 × 1 = 720 So, the odds are 2,763,633,600 ÷ 720 = 3,838,380 to one.

If I put $50 on one lottery, the odds of winning are 3,838,380 ÷ 50 = 76767.6 to one. That is the easy part.

The hard part is calculating the odds of winning if I put $1 on 50 lotteries. To do this we have to convert to probabilities. A probability is a number between 1 and 0. 1 means a perfect chance, 0 means no chance at all. The probability of winning the lottery with one dollar is 1 ÷ 3,838,380 = 0.0000002605... in other words, very small. The probability of winning the lottery with 50 dollars is 1 ÷ 76767.6 = 0.0000130263288...

Just for your information, the probability of winning twice in a row is this number squared which is 0.00000000000006. This is not very useful information except that we can use this formula in reverse to get what we want. The cool thing about probabilities, as opposed to odds, is that the probability of winning is one minus the probability of losing. This leads us to a way of calculating what we want by using a double negative. The probability of winning at least once in 50 tries is the same as the probability of not losing 50 times in a row.

The probability of losing is 1 - 0.0000002605... = 0.9999997394...

The probability of losing 50 times in a row is 0.9999997394... to the 50th power = 0.99998697...

The probability of not losing 50 times in a row is 1 - 0.99998697... = 0.0000130262457...

So, since the probability of winning at least once in 50 tries is 0.0000130262457... and the probability of winning one lottery with 50 dollars is 0.0000130263288... The odds are very very slightly more favorable by playing all 50 dollars in one lottery than spreading it out among 50 lotteries.

Another way to look at lotteries is with expected returns. This gives another slight advantage to putting 50 dollars on a single lottery when you wait for the jackpot to exceed the odds of winning. As stated in the Glossary, under the category of Sucker Bet, the formula is: EXPECTED RETURN = POTENTIAL WINNING * PROBABILITY OF WINNING - POTENTIAL COST * PROBABILITY OF LOSING.

Given the lottery above, if the jackpot starts at $1 million, the potential winning is 1 million, the probability of winning is 0.0000002605..., the potential cost is $1, and as many people have mentioned in response, the probability of losing is 1!

The expected return is 0.2605 - 1 = -0.7395, or a loss of 74 cents. What is not realized is that for every dollar put into the lottery at least 50 cents go to the state general fund, with some states taking as many as 70 cents on the dollar. A better expected return can be found by waiting for the jackpot to exceed $3,838,380. A four million dollar jackpot would have an expected gain of 4 cents.

Still, with the likelihood of winning being so small, you are better off burning your money, or giving it to charity. Lottery is a voluntary tax. I give enough to taxes, thank you very much. If my money is going to benefit someone else, it will be someone of my choosing.