How To Calculate Winning Odds in California Lottery

Ever wonder how to calculate winning odds of lottery games? The winning odds of the top prize of Fantasy 5 in California Lottery are 1 in 575,757. The winnings odds of the top prize of SuperLOTTO plus are 1 in 41,416,353. The winnings odds of the top prize of Mega Millions are 1 in 175,711,534. In this post, we show how to calculate the odds for these games in the California Lottery. The calculation is an excellent combinatorial exercise as well as in calculating hypergeometric probability.

All figures and data are obtained from the California Lottery.

Update, April 27, 2017. The calculation in this post assumes certain background knowledge on combination and the multiplication principle (not explained here). For any reader who would like to further understand how lottery odds are calculated, see this blog post on Powerball. It is a self contained step by step explanation at the basic level on how to calculate winning odds in the Powerball game.

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Fantasy 5

The following figures show a playslip and a sample ticket for the game of Fantasy 5.

Figure 1

Figure 2

In the game of Fantasy 5, the player chooses 5 numbers from 1 to 39. If all 5 chosen numbers match the 5 winning numbers, the player wins the top prize which starts at $50,000 and can go up to $500,000 or more. The odds of winning the top prize are 1 in 575,757. There are lower tier prizes that are easier to win but with much lower winning amounts. The following figure shows the prize categories and the winning odds of Fantasy 5.

Figure 3

All 5 of 5
In matching the player’s chosen numbers with the winning numbers, the order of the numbers do not matter. Thus in the calculation of odds, we use combination rather than permutation. Thus we have:

\displaystyle (1) \ \ \ \ \ \binom{39}{5}=\frac{39!}{5! \ (39-5)!}=575757

Based on (1), the odds of matching all 5 winning numbers is 1 in 575,757 (the odds of winning the top prize).

Any 4 of 5
To match 4 out of 5 winning numbers, 4 of the player’s chosen numbers are winning numbers and 1 of the player’s chosen numbers is from the non-winning numbers (34 of them). Thus the probability of matching 4 out of 5 winning numbers is:

\displaystyle (2) \ \ \ \ \ \frac{\displaystyle \binom{5}{4} \ \binom{34}{1}}{\displaystyle \binom{39}{5}}=\frac{5 \times 34}{575757}=\frac{1}{3386.8} \ \ \text{(1 out of 3,387)}

Any 3 of 5
To find the odds for matching 3 out of 5 winning numbers, we need to find the probability that 3 of the player’s chosen numbers are from the 5 winning numbers and 2 of the selected numbers are from the 34 non-winning numbers. Thus we have:

\displaystyle (3) \ \ \ \ \ \frac{\displaystyle \binom{5}{3} \ \binom{34}{2}}{\displaystyle \binom{39}{5}}=\frac{10 \times 561}{575757}=\frac{1}{102.63} \ \ \text{(1 out of 103)}

Any 2 of 5
Similarly, the following shows how to calculate the odds of matching 2 out of 5 winning numbers:

\displaystyle (4) \ \ \ \ \ \frac{\displaystyle \binom{5}{2} \ \binom{34}{3}}{\displaystyle \binom{39}{5}}=\frac{10 \times 5984}{575757}=\frac{1}{9.6216} \ \ \text{(1 out of 10)}

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SuperLOTTO Plus

Here are the pictures of a playslip and a sample ticket of the game of SuperLOTTO Plus.

Figure 4

Figure 5

Based on the playslip (Figure 4), the player chooses 5 numbers from 1 to 47. The player also chooses an additional number called a Mega number from 1 to 27. To win the top prize, there must be a match between the player’s 5 selections and the 5 winning numbers as well as a match between the player’s Mega number and the winning Mega number (All 5 of 5 and Mega in Figure 6 below).

Figure 6

All 5 of 5 and Mega
To find the odds of the match of “All 5 of 5 and Mega”, the total number of possibilities is obtained by choosing 5 numbers from 47 numbers and choose 1 number from 27 numbers. We have:

\displaystyle (5) \ \ \ \ \ \binom{47}{5} \times \binom{27}{1}=41,416,353

Thus the odds of matching “All 5 of 5 and Mega” are 1 in 41,416,353.

Any 5 of 5
To find the odds of matching “All 5 of 5” (i.e. the player’s 5 selections match the 5 winning numbers but no match with the Mega winning number), we need to choose 5 numbers from the 5 winning numbers, choose 0 numbers from the 42 non-winning numbers, choose 0 numbers from the 1 Mega winning number and choose 1 number from the 26 non-Mega winning numbers. This may seem overly precise, but will make it easier to the subsequent derivations. We have:

\displaystyle \begin{aligned}(6) \ \ \ \ \   \frac{\displaystyle \binom{5}{5} \ \binom{42}{0} \ \binom{1}{0} \ \binom{26}{1}}{\displaystyle \binom{47}{5} \times \binom{27}{1}}&=\frac{1 \times 1 \times 1 \times 26}{41416353} \\&=\frac{1}{1592936.654} \\&\text{ } \\&=\text{1 out of 1,592,937}  \end{aligned}

Any 4 of 5 and Mega
To calculate the odds for matching “any 4 of 5 and Mega”, we need to choose 4 out of 5 winning numbers, choose 1 out of 42 non-winning numbers, choose 1 out of 1 Mega winning number, and choose 0 out of 26 non-winning Mega numbers. We have:

\displaystyle \begin{aligned}(7) \ \ \ \ \   \frac{\displaystyle \binom{5}{4} \ \binom{42}{1} \ \binom{1}{1} \ \binom{26}{0}}{\displaystyle \binom{47}{5} \times \binom{27}{1}}&=\frac{5 \times 42 \times 1 \times 1}{41416353} \\&=\frac{1}{197220.7286} \\&\text{ } \\&=\text{1 out of 197,221}  \end{aligned}

Any 4 of 5
To calculate the odds for matching “any 4 of 5” (no match for Mega number), we need to choose 4 out of 5 winning numbers, choose 1 out of 42 non-winning numbers, choose 0 out of 1 Mega winning number, and choose 1 out of 26 non-winning Mega numbers. We have:

\displaystyle \begin{aligned}(8) \ \ \ \ \   \frac{\displaystyle \binom{5}{4} \ \binom{42}{1} \ \binom{1}{0} \ \binom{26}{1}}{\displaystyle \binom{47}{5} \times \binom{27}{1}}&=\frac{5 \times 42 \times 1 \times 26}{41416353} \\&=\frac{1}{7585.412637} \\&\text{ } \\&=\text{1 out of 7,585}  \end{aligned}

Any 3 of 5 and Mega
To calculate the odds for matching “any 3 of 5 and Mega”, we need to choose 3 out of 5 winning numbers, choose 2 out of 42 non-winning numbers, choose 1 out of 1 Mega winning number, and choose 0 out of 26 non-winning Mega numbers. We have:

\displaystyle \begin{aligned}(9) \ \ \ \ \   \frac{\displaystyle \binom{5}{3} \ \binom{42}{2} \ \binom{1}{1} \ \binom{26}{0}}{\displaystyle \binom{47}{5} \times \binom{27}{1}}&=\frac{10 \times 861 \times 1 \times 1}{41416353} \\&=\frac{1}{4810.261672} \\&\text{ } \\&=\text{1 out of 4,810}  \end{aligned}

The rest of the calculations for SuperLOTTO Plus should be routine. It is a matter to deciding how many to choose from the 5 winning numbers, how many to choose from the 42 non-winning numbers as well as how many to choose from the 1 winning Mega number and how many to choose from the 26 non-winning Mega numbers.

Any 3 of 5
\displaystyle \begin{aligned}(10) \ \ \ \ \   \frac{\displaystyle \binom{5}{3} \ \binom{42}{2} \ \binom{1}{0} \ \binom{26}{1}}{\displaystyle \binom{47}{5} \times \binom{27}{1}}&=\frac{10 \times 861 \times 1 \times 26}{41416353} \\&=\frac{1}{185.0100643} \\&\text{ } \\&=\text{1 out of 185}  \end{aligned}

Any 2 of 5 and Mega
\displaystyle \begin{aligned}(11) \ \ \ \ \   \frac{\displaystyle \binom{5}{2} \ \binom{42}{3} \ \binom{1}{1} \ \binom{26}{0}}{\displaystyle \binom{47}{5} \times \binom{27}{1}}&=\frac{10 \times 11480 \times 1 \times 1}{41416353} \\&=\frac{1}{360.7696254} \\&\text{ } \\&=\text{1 out of 361}  \end{aligned}

Any 1 of 5 and Mega
\displaystyle \begin{aligned}(12) \ \ \ \ \   \frac{\displaystyle \binom{5}{1} \ \binom{42}{4} \ \binom{1}{1} \ \binom{26}{0}}{\displaystyle \binom{47}{5} \times \binom{27}{1}}&=\frac{5 \times 111930 \times 1 \times 1}{41416353} \\&=\frac{1}{74.00402573} \\&\text{ } \\&=\text{1 out of 74}  \end{aligned}

None of 5 only Mega
\displaystyle \begin{aligned}(13) \ \ \ \ \   \frac{\displaystyle \binom{5}{0} \ \binom{42}{5} \ \binom{1}{1} \ \binom{26}{0}}{\displaystyle \binom{47}{5} \times \binom{27}{1}}&=\frac{1 \times 850668 \times 1 \times 1}{41416353} \\&=\frac{1}{48.68685903} \\&\text{ } \\&=\text{1 out of 49}  \end{aligned}

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Mega Millions

The following are a playslip, a sample ticket and the winning odds of the game of Mega Millions.

Figure 7

Figure 8

Figure 9

Based on the playslip (Figure 7), the player chooses 5 numbers from 1 to 56. The player also chooses an additional number called a Mega number from 1 to 46. To win the top prize, there must be a match between the player’s 5 selections and the 5 winning numbers as well as a match between the player’s Mega number and the winning Mega number. The calculation of the odds indicated in Figure 9 are left as exercises.

Picking Two Types of Binomial Trials

We motivate the discussion with the following example. The notation W \sim \text{binom}(n,p) denotes the statement that W has a binomial distribution with parameters n and p. In other words, W is the number of successes in a sequence of n independent Bernoulli trials where p is the probability of success in each trial.

Example 1
Suppose that a student took two multiple choice quizzes in a course for probability and statistics. Each quiz has 5 questions. Each question has 4 choices and only one of the choices is correct. Suppose that the student answered all the questions by pure guessing. Furthermore, the two quizzes are independent (i.e. results of one quiz will not affect the results of the other quiz). Let X be the number of correct answers in the first quiz and Y be the number of correct answers in the second quiz. Suppose the student was told by the instructor that she had a total of 4 correct answers in these two quizzes. What is the probability that she had 3 correct answers in the first quiz?

On the face of it, the example is all about binomial distribution. Both X and Y are binomial distributions (both \sim \text{binom}(5,\frac{1}{4})). The sum X+Y is also a binomial distribution (\sim \text{binom}(10,\frac{1}{4})). The question that is being asked is a conditional probability, i.e., P(X=3 \lvert X+Y=4). Surprisingly, this conditional probability can be computed using the hypergeometric distribution. One can always work this problem from first principle using binomial distributions. As discussed below, for a problem such as Example 1, it is always possible to replace the binomial distributions using a thought process involving the hypergeometric distribution.

Here’s how to think about the problem. This student took the two quizzes and was given the news by the instructor that she had 4 correct answers in total. She now wonders what the probability of having 3 correct answers in the first quiz is. The thought process is this. She is to pick 4 questions from 10 questions (5 of them are from Quiz 1 and 5 of them are from Quiz 2). So she is picking 4 objects from a group of two distinct types of objects. This is akin to reaching into a jar that has 5 red balls and 5 blue balls and pick 4 balls without replacement. What is the probability of picking 3 red balls and 1 blue ball? The probability just described is from a hypergeometric distribution. The following shows the calculation.

\displaystyle (1) \ \ \ \ P(X=3 \lvert X+Y=4)=\frac{\displaystyle \binom{5}{3} \ \binom{5}{1}}{\displaystyle \binom{10}{4}}=\frac{50}{210}

We will show below why this works. Before we do that, let’s describe the above thought process. Whenever you have two independent binomial distributions X and Y with the same probability of success p (the number of trials does not have to be the same), the conditional distribution X \lvert X+Y=a is a hypergeometric distribution. Interestingly, the probability of success p has no bearing on this observation. For Example 1, we have the following calculation.

\displaystyle (2a) \ \ \ \ P(X=0 \lvert X+Y=4)=\frac{\displaystyle \binom{5}{0} \ \binom{5}{4}}{\displaystyle \binom{10}{4}}=\frac{5}{210}

\displaystyle (2b) \ \ \ \ P(X=1 \lvert X+Y=4)=\frac{\displaystyle \binom{5}{1} \ \binom{5}{3}}{\displaystyle \binom{10}{4}}=\frac{50}{210}

\displaystyle (2c) \ \ \ \ P(X=2 \lvert X+Y=4)=\frac{\displaystyle \binom{5}{2} \ \binom{5}{2}}{\displaystyle \binom{10}{4}}=\frac{100}{210}

\displaystyle (2d) \ \ \ \ P(X=3 \lvert X+Y=4)=\frac{\displaystyle \binom{5}{3} \ \binom{5}{1}}{\displaystyle \binom{10}{4}}=\frac{50}{210}

\displaystyle (2e) \ \ \ \ P(X=4 \lvert X+Y=4)=\frac{\displaystyle \binom{5}{4} \ \binom{5}{0}}{\displaystyle \binom{10}{4}}=\frac{5}{210}

Interestingly, the conditional mean E(X \lvert X+Y=4)=2, while the unconditional mean E(X)=5 \times \frac{1}{4}=1.25. The fact that the conditional mean is higher is not surprising. The student was lucky enough to have obtained 4 correct answers by guessing. Given this, she had a greater chance of doing better on the first quiz.

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Why This Works

Suppose X \sim \text{binom}(5,p) and Y \sim \text{binom}(5,p) and they are independent. The joint distribution of X and Y has 36 points in the sample space. See the following diagram.

Figure 1

The probability attached to each point is

\displaystyle \begin{aligned}(3) \ \ \ \  P(X=x,Y=y)&=P(X=x) \times P(Y=y) \\&=\binom{5}{x} p^x (1-p)^{5-x} \times \binom{5}{y} p^y (1-p)^{5-y}  \end{aligned}

where x=0,1,2,3,4,5 and y=0,1,2,3,4,5.

The conditional probability P(X=k \lvert X+Y=4) involves 5 points as indicated in the following diagram.

Figure 2

The conditional probability P(X=k \lvert X+Y=4) is simply the probability of one of the 5 sample points as a fraction of the sum total of the 5 sample points encircled in the above diagram. The following is the sum total of the probabilities of the 5 points indicated in Figure 2.

\displaystyle \begin{aligned}(4) \ \ \ \  P(X+Y=4)&=P(X=0) \times P(Y=4)+P(X=1) \times P(Y=3)\\&\ \ \ \ +P(X=2) \times P(Y=3)+P(X=3) \times P(Y=2)\\&\ \ \ \ +P(X=4) \times P(Y=0)  \end{aligned}

We can plug (3) into (4) and work out the calculation. But (4) is actually equivalent to the following because X+Y \sim \text{binom}(10,p).

\displaystyle (5) \ \ \ \ P(X+Y=4)=\ \binom{10}{4} p^4 \ (1-p)^{6}

As stated earlier, the conditional probability P(X=k \lvert X+Y=4) is simply the probability of one of the 5 sample points as a fraction of the sum total of the 5 sample points encircled in Figure 2. Thus we have:

\displaystyle \begin{aligned}(6) \ \ \ \  P(X=k \lvert X+Y=4)&=\frac{P(X=k) \times P(Y=4-k)}{P(X+Y=4)} \\&=\frac{\displaystyle \binom{5}{k} p^k (1-p)^{5-k} \times \binom{5}{4-k} p^{4-k} (1-p)^{5-(4-k)}}{\displaystyle  \binom{10}{4} p^4 \ (1-p)^{6}}   \end{aligned}

With the terms involving p and 1-p cancel out, we have:

\displaystyle (7) \ \ \ \  P(X=k \lvert X+Y=4)=\frac{\displaystyle \binom{5}{k} \times \binom{5}{4-k}}{\displaystyle  \binom{10}{4}}

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Summary

Suppose X \sim \text{binom}(N,p) and Y \sim \text{binom}(M,p) and they are independent. Then X+Y is also a binomial distribution, i.e., \sim \text{binom}(N+M,p). Suppose that both binomial experiments \text{binom}(N,p) and \text{binom}(M,p) have been performed and it is known that there are a successes in total. Then X \lvert X+Y=a has a hypergeometric distribution.

\displaystyle (8) \ \ \ \  P(X=k \lvert X+Y=a)=\frac{\displaystyle \binom{N}{k} \times \binom{M}{a-k}}{\displaystyle  \binom{N+M}{a}}

where k=0,1,2,3,\cdots,\text{min}(N,a).

As discussed earlier, think of the N trials in \text{binom}(N,p) as red balls and think of the M trials in \text{binom}(M,p) as blue balls in a jar. Think of the a successes as the number of balls you are about to draw from the jar. So you reach into the jar and select a balls without replacement. The calculation in (8) gives the probability that you select k red balls and a-k blue balls.

The probability of success p in the two binomial distributions have no bearing on the result since it gets canceled out in the derivation. One can always work a problem like Example 1 using first principle. Once the thought process using hypergeometric distribution is understood, it is a great way to solve this problem, that is, you can by pass the binomial distributions and go straight to the hypergeometric distribution.

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Additional Practice
Practice problems are found in the following blog post.

How to pick binomial trials