Chances of winning the lottery (lottery odds)

The probability of winning the lottery depends on the number of possible combinations of ball draws, and we will now learn how to calculate them ourselves. For those who don't want to calculate themselves, there's an online calculator at the end.

Probability

— is the degree (relative measure, quantitative assessment) of the possibility of an event occurring.

Let's start with something simple, we have five balls:

What is the probability of guessing one ball out of five? It is equal to \(\frac{1}{5}\) , as there are only five possible combinations for this set of numbers: either 5 , 3 , 2 , 4 or 1 .

For further convenience, let's denote our lotteries as « \(k\text{ из }n\) », and when necessary, we'll substitute the corresponding numbers.

Let's complicate the rules of our lottery - to win, you need to guess «2 out of 5» ( \(k = 2, n = 5\) ). Now the chance of guessing is \(\frac{1}{10}\) , as there are ten possible combinations, here they are:

It's important to note that for winning the lottery, the order of numbers in each combination doesn't matter.

In probability theory, the aforementioned five balls are actually a set of numbers from 1 to 5. A set is denoted by curly braces { }, and each individual combination is called a combination.

In combinatorics, a combination of k out of n elements refers to a combination containing k elements chosen from a set containing n different elements.

In combinations, the order of elements is not considered, \(\{1, 2\}\) and \(\{2, 1\}\) are considered the same.

Now we can express all this mathematically:

We have a set of 5 balls \(\{1, 2, 3, 4, 5\}\) . And there are 10 combinations that can be formed from 5 choose 2 balls:

The number of all combinations of n elements choose k each is denoted as \(C_{n}^{k}\) (from the initial letter of the French word «combinasion», which means «combination») and is read as «the number of combinations of n elements choose k». In our case, \(C_{5}^{2}\) — the number of combinations of 5 choose 2 is equal to 10.

The number of combinations

The number of combinations is calculated by the formula:

\(C_{n}^{k} = \frac{n!}{k!(n-k)!}\)

\(n!\) and \(k!\) — are the factorials of the respective numbers \(n\) and \(k\) . The factorial of a natural number \(n\) is the product of all natural numbers from 1 to \(n\) inclusive. For example, the factorial of the number 5 is \(5! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120\) .

Let's check our result for the 2 out of 5 lottery:

\(C_{5}^{2} = \frac{5!}{2!(5-2)!}= \frac{5!}{2!\cdot3!} = \frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5}{1 \cdot 2 \cdot 1 \cdot 2 \cdot 3}\)

Look, we can simplify the dividend and divisor \((n-k)!\) I highlighted it in parentheses to make it clearer:

\(\frac{(1 \cdot 2 \cdot 3) \cdot 4 \cdot 5}{1 \cdot 2 \cdot ( 1 \cdot 2 \cdot 3)} = \frac{1 \cdot 2 \cdot 3}{1 \cdot 2 \cdot 3} \cdot \frac{ 4 \cdot 5}{1 \cdot 2} = \frac{20}{2} = 10\)

Note that after we have simplified the dividend and divisor, we have two numbers left in both the dividend and divisor, specifically \(k\) numbers. In the dividend, it is the product of the two largest numbers from \(n\) , and in the divisor, it is the factorial of the number \(k\) . And if you want to calculate the probability of winning, you don't need to calculate the factorials completely, just multiply the \(k\) largest elements from \(n\) and divide by the factorial of \(k\) .

Let's calculate the number of combinations for the «6 out of 45» lottery:

\(C_{45}^{6} = \frac{45!}{6!(45-6)!} = \frac{45 \cdot 44 \cdot 43 \cdot 42 \cdot 41 \cdot 40}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = \frac{5,864,443,200}{720} = 8,145,060\)

The entire set of combinations is a complete system. If you buy tickets with all combinations, you will guaranteed win.

Probability of Winning

Now let's move on to the probability of winning. If you buy a ticket for the «6 out of 45» lottery with one combination, your chances are 1 in 8,145,060. If you take 2 tickets with different combinations, your chances are 2 in 8,145,060 or 1 in 4,072,530. If you take 10 tickets, but write down the same combination everywhere, your chances are again 1 in 8,145,060. Thus, probability is the ratio of the number of your unique combinations to the total number of combinations.

If you are playing a lottery where you need to correctly guess numbers in two game fields, for example, in the Powerball lottery «5 out of 69 + 1 out of 26», then you need to multiply the number of combinations for «5 out of 69» by «1 out of 26».

\(C_{69}^{5} = \frac{69!}{5!(69-5)!} = \frac{69 \cdot 68 \cdot 67 \cdot 66 \cdot 65}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = \frac{1,348,621,560}{120} = 11,238,513\)

\(C_{26}^{1} = \frac{26!}{1!(26-1)!} = \frac{26}{1} = 26\)

\(11,238,513 \cdot 26 = 292,201,338\)

«4 из 20»

In the «4 out of 20» lottery, to win the jackpot, you need to correctly guess «4 out of 20» in two fields. Let's calculate the number of combinations for one field:

\(C_{20}^{4} = \frac{20!}{4!(20-4)!} = \frac{20 \cdot 19 \cdot 18 \cdot 17}{4 \cdot 3 \cdot 2 \cdot 1} = \frac{116,280}{4,845} = 4,845\)

We get 4,845 combinations, so the probability of guessing «4 out of 20» is 1 in 4,845. But since we need to guess twice, we multiply the probabilities to get the number of combinations for two fields:

\(4,845 \cdot 4,845 = 23,474,025\)

As we can see, the probability of winning in «4 out of 20» is lower than in «6 out of 45», 1 in 23 million versus 1 in 8 million.

You can also look at the probability of winning the lottery when using expanded bets .