What is the probability of winning $100,000?

Question

In a particular lottery game, a player picks 5 numbers from 1 to 51. If 4 of the 5 numbers match those drawn, the player wins $100,000. What is the probability of winning $100,000?

Answer 1

Let's first consider the case where exactly 4 of the 5 numbers match. We also assume that all combinations of 5 numbers are equally likely.

We can start with finding the total number of combinations that can be drawn. This is fairly straightforward as we are choosing 5 numbers out of 51. So, the total number of combinations is 51 choose 5 (you can plug in the numbers to the combination formula and simplify).

We can first divide up the numbers into winners (5) and losers (51-5 = 46). We want to figure out how many winning combinations are there. Since we want exactly 4 matches, for the winning numbers, it would be 5 choose 4. Since we also want exactly 1 mismatch, for the losing numbers, it would be 46 choose 1, or just 46. We can multiply 5 choose 4 with 46 to get the number of winning combinations.

Since all 5 number combinations are equally likely, we can take the number of winning combinations and divide by the total number of combinations to get the probability.

In case choosing all 5 winning numbers is an option, there is only 1 way to choose all 5 winning numbers. You can use the same logic as the preceding paragraph to figure out this probability.