Game rules:

All players have to choose three numbered balls out of ten. e.g. 1, 2 and 3 or 5, 7 and 10.

No same numbers are repeatedly chosen.

After each ball is drawn, it cannot be put back with the other balls.

Only three balls will be drawn out of the ten numbered balls.

Each selection of three balls costs each player one dollar.

Question: What is the probability of drawing your favorite three balls out of the ten?

Do not worry about the complex mathematics involved here. Once you have read this article, you will find it simple step by step to understand the principle behind. As most students know, the probability of drawing the three chosen numbers out of ten is 1/10 × 1/9 × 1/8. Hence the answer should be 1/720. If the jackpot is $1000, you may think that it is worthwhile to bet across the board and you will pay $720 to try your luck. (Net earnings: $1000 – $720 = $280) However, it is a serious mistake and you have paid too much. You could have saved some money to bet next time or buy some goodies you like.

Before getting to the bottom of it, we take an easy simulation to understand why it is wrong to bet in such a silly way. The probability of drawing three different balls out of five with the same game rules SHOULD be 1/5 × 1/4 × 1/3. It means 1/60 and there SHOULD be 60 chances to draw the three balls. However, we must consider how the three balls come out. It is supposed that 1, 2, and 3 were randomly drawn out of the 5 balls. The three balls (1, 2 and 3) can show different patterns of priorities. For example,

Therefore, there ARE six chances to win the first prize out of the sixty chances and the probability IS six out of sixty (6/60). We can see that it is changed from 1/60 to 6/60 and the exact probability of drawing three different balls out of five is 1/10. If you would like to bet across the board again, you only need to pay $10 other than $60. That is why you could have saved some money ($60 – $10 = $50) to bet next time as mentioned above. Of course, I am not encouraging you to gamble. I just instruct readers how to calculate wisely.

Let’s go back to the ten balls. There are the same 6 patterns of the three drawn balls out of the ten too. When 6 patterns are over the 720 chances (i.e. 6/720), the correct probability of drawing your favorite three balls out of the ten is 1/120 rather than 1/720. Net savings of the bet is $600 when $120 is subtracted from $720. ($720 – $120 = $600)

Up to here, you may want to ask me back a question. How can we count the total number of patterns? You are smart to have this problem. When there are 3 balls that have to be drawn, the total patterns are 6 because 1 × 2 × 3 = 6. When there are 4 balls that have to be drawn, the total patterns are 24 because 1 × 2 × 3 × 4 = 24. When there are 5 balls that have to be drawn, the total patterns are 120 because 1 × 2 × 3 × 4 × 5 = 120. Can you see the regular way to figure out the number of total patterns with a certain number of drawn balls now?

May I have your attention, please? It is your show time at this place and at this time. Could you tell me the probability of winning the first prize when 6 numbered balls are randomly drawn one after one without putting them back out of 50 balls? The first one to answer me correctly has the probability to mooch a free air ticket to Las Vegas. Wow!