In a typical large lottery, the numbers that can be picked fall within a specific range. A common range is 1 to 50.

This means the odds that someone will pick the first number correctly on it's own is 1 in 50. Since you cannot choose the same number again, your next choice will have odds of 1 on 49 since there are only 49 numbers left to choose from. By the third pick, the individual odds will be 1 in 48 that you pick the correct number.

As there essentially fractions we can simply multiply these odds together to get the cumulative odds after 3 picks.

Pick 1 - 1/50
Pick 2 - 1/49 * 1/50 = 1/2450
Pick 3 - 1/48 * 1/49 * 1/50 = 1/117600

Since this does not account for the order that you pick the numbers, only that you do not reuse numbers, we must also account for the fact that there are 6 (3 factorial) possible ways to represent the same 3 numbers. That means we need to remove that number of possible outcomes from our final odds by dividing the product of the above picks by this factorial.

You have a 1 in 19,600 chance of picking the first 3 numbers correctly, assuming the numbers range from 1 to 50.
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