程阳：1次50注，50次1注，哪个中奖机会高？
以前以为中国民众愚钝，现在连数学家也犯错了，才感到事态严重。

Is it better to play 50 dollars one week,or one dollar for fifty weeks?

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.

Expected Return

Another way to look at lotteries is with expected returns. This gives another sleight 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.

The Best way to play the Lottery

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.

Power Ball

On July 29, 1998 the 20 state Power Ball lottery gave away a record $250,000,000 or a quarter of a billion dollars. The odds of winning are around 80 million to 1 per ticket. Even if you decide to take it in a lump sum, it is worth $130 million.

Such gigantic sums of money are tempting lots of people. People are using their rent checks and paychecks to buy hundreds of tickets in an attempt to better their odds.

Sure every dollar you spend increases your chances of winning, but each increase is minor. Spending $2 increases your chances from 80 million:1 to 40 million:1.

Yeah you doubled your chances, but as we learned in elementary 2 times 0 is 0, so it follows that 2 times almost 0 is also almost 0.

The odds are calculated as follows:

There are 49 white balls in which 5 are chosen, the order does not matter. Then one of 42 red balls is chosen. Thus we have the formula:

Here is a better way to play the lottery, write down 5 numbers between 1 and 49, then pick a Powerball number between 1 and 42 (you can reuse any of the previous 5 or pick another number). Don't buy a ticket, then when the drawing is held, and your numbers don't match, you can congratulate yourself for saving a dollar.