6904

Saw this article in the papers today. My eyes popped out. Apparently 6904 won the top 2 prizes in 4D. The Singapore Pools spokesman said that the chance of this happening was 1 in 100 million. This is preposterous. If Singapore Pools people are so bad in calculating probability, I would want to be betting against them for football results. Actually I do that, and on average, I’m more or less even.

Now technically, he’s right. The chances that 6904 will win the top prize is indeed 1 in 100 million. The answer is correct but the question is wrong. The question should have been “what are the chances that the first 2 numbers are the same?" Then it is a less astronomical 1 in 10000. Considering that there are 156 draws in a year, on average this means once in 66 years. A rare event, to be sure, but not that rare.

Of course, this method of calculation also applies to “what are the chances that the same number will win both x and y?” Where x and y are 2 different prizes. So if x and y are 2 consecutive first prizes, then the answer is the same: 1 in 10000, once in 66 years.

Then the question arises: what is the chance that the first prize and another prize will be the same? This is not easy to calculate. But it is easy to calculate what are the chances that all the other winners are different from the first prize. So if the first prize is ABCD, then the probability that the 2nd prize is not ABCE is 0.9999, same as the 3rd prize, each starter prize, each consolation prize. The probability that none of the prizes is same as the first prize is therefore (0.9999 ^ 22) and the complement set of that is 0.0022, or 1 in 455. So every 3 years, you will find the 1st prize having the same number as 1 other prize.


The spokesman says “the same four-digit number coming up twice is not unprecedented, but usually, it appears in the starter or consolation category, not as a double first-second prize whammy.” Now this is not quite as stupid as his other statement. What are the chances that at least 2 different prizes have the same number? The probability of the second prize being different from the first is 0.9999. The probability of the 3rd prize being different from 1 and 2, if 1 and 2 are different, is 0.9998. Etc etc etc. So for the 23 prizes, (0.9999 * 0.9998 * … * 0.9978) is 0.975. 1 – 0.975 is 0.025, which is 1 in 40, which means it happens roughly every 13 weeks or so.