Last updated on February 8, 2021

Probability and odds are two related concepts, but they are not mathematically equivalent.

Therefore, discussing probability and odds must include their difference in meaning and in scale.

Some think it matters not what term is used, as long as you get the gist.

However, it could lead to flawed decision making and incorrect estimates of chance if the exact term gets jumbled in a wrong context.

Distinguishing probability from odds

The inappropriate swapping of the terms “probability” and “odds” is widespread in many state lottery websites. If you lack the insight to perceive this, you might end up making the wrong decisions when playing.

It is, therefore, necessary to know the difference between the two related mathematical concepts. In lottery games, for example, knowing the difference between probability and odds could help you decide which combination to play.

Disclaimer: I am not saying that the computations of odds and probabilities on state lottery websites are wrong. The purpose of this article is to simply set a clear definition and context for probability and odds.

Probability refers to the ratio of the number of times an outcome could occur compared to the number of all possible outcomes.

In our previous posts, we use the formula below for probability.

In a lottery game, the probability of winning offered by one combination you mark on your playslip is one over the total number of possible combinations.

For example, you bought a ticket for a 6/47 game for the combination 1-2-3-4-5-6. In order to bring home the jackpot, you need to exactly match the winning combination.

A 6/47 game has a total possible combinations of 10,737,573. Therefore, the probability is 1/10,737,573. A common way of expressing probability in spoken language is x in y.

Hence, the probability to win in a 6/47 game with 1-2-3-4-5-6 combination is 1 in 10,737,573.

Odds also refer to a ratio. This time, however, it is the ratio of favorable outcomes compared to unfavorable outcomes.

Odds compare the number of ways an event can occur with the number of ways the event cannot occur.

We have been using the formula below to compute for odds.

We aptly refer to odds as the ratio of success to failure because the odds favoring your winning the lottery is the number of success over the number of failures.

Using the formulas for odds, we can compute for the odds as 1/ (10,737,573 – 1) or 1/10,737,572.

In our other posts, we express odds or ratio of success to failure as x to y. Hence, the odds for winning in a 6/47 lotto game with the combination 1-2-3-4-5-6 is 1 to 10,737,572.

Others also denote odds as x: y so we can also write 1 to 10,737,572 as 1: 10,737,572.

This is just for the jackpot prize.

We may also calculate the second division prize for matching 5 out of 6 balls.

C(6,5)= Number of ways to match 5 balls (6 ways to happen)

C(41,1) = The sixth ball must be one of the remaining 41 balls that were not drawn (41 ways this can happen)

(6 * 41) = 246 ways you can match 5 of 6

We have to minus the 246 from the total number of combinations. Therefore, there are 10,737,327 ways to fail.

10,737,573 – 246 = 10,737,327

With this, the expression of odds should be:

Odds (5 of 6) = 246 / 10,737,327

or

Odds (5 of 6) = 1 : 43,648

Clearly, it shouldn’t be 1 : 43,649 as shown in the Official Michigan Lotto 47 odds table shown below.

The same can be said for other minor prize divisions.

Confusing information about odds and probability in lotteries is widespread. In the following discussion, you will see that there are only at least two state lotteries that hit the correct mark in declaring the probability of winning for the games they offer.

Apparently, 10 other state lotteries do not show the correct information that players need to know. These are only a few examples, but expect to see more lotteries with confusing odds and probability details.

Make sure that you have the proper knowledge to distinguish odds from probability and vice versa. This way, you will be prepared to realize for yourself what you must do when you see the inaccurate information.

Massachusetts Lottery

There are only at least two state lotteries that provide information to their players based on how we recognize and use probability and odds.

Among them is Massachusetts Lottery.

This is a table for 6/69 Megabucks Doubler of Massachusetts Lottery.

The information provided by the Massachusetts Lottery to its patrons coincides with how we explain probability and odds to discerning readers. You see from the table above that the probability to win the jackpot by matching 6 out of 6 numbers is 1 in 13,983,816.

This is also how Massachusetts Lottery provided players with the crucial probability information for its other draw games. Expect to see a similar representation of probability for Mass Cash, Lucky for Life, Powerball and Mega Millions.

The probability to win the jackpot in Mass Cash is 1 in 324,632.

In Lucky for Life, you could win $7,000 a WEEK for LIFE! by matching the 5 numbers and the Lucky Balls at a probability of 1 in 30,821,472.

Confusion could arise looking at the winning odds from Powerball website and the winning probability from Massachusetts Powerball web page. The Powerball website notes that the odds to win the grand prize are 1 in 292,201,338.

The probability of winning the game from the Massachusetts webpage aligns more with our understanding of probability. The “1 in 292,201,338” is not the odds, but the probability to win.

A similar situation exists for Mega Millions. The Massachusetts web page for Mega Million depicts the probability to win this game as 1 in 302,575,350.

Massachusetts is not alone in presenting probability this way. There is also Pennsylvania Lottery.

Pennsylvania Lottery

Pennsylvania Lottery, meanwhile, does not claim outright that the information it provides is odds or probability. See the image below to see what I mean.

Instead of stating directly whether it is odds or probability, Pennsylvania Lottery uses “chances of winning”.

Incidentally, probability also refers to the number reflecting the chance that a particular event will occur. It is also valid to call probability as chance.

Hence, the way Pennsylvania Lottery presented chances of winning is the same as saying probability of winning. From the information in the table, the probability or chance to win the jackpot in the Pennsylvania Lottery Treasure Hunt is 1 in 142,506.

You could also view similar presentation of probability for Pennsylvania Lottery’s other draw games like Cash4Life, Cash 5, Powerball and Mega Millions.

It is unfortunate that other state lotteries do not have the same manner of imparting knowledge to its regulars on probability and odds. In this day and age of technology, one must be insightful when reading and accepting any presented information. This will help eliminate chances of deciding incorrectly.

Ohio Lottery

Take, for instance, this table for Ohio Lottery Classic Lotto.

Notice that this Classic Lotto from Ohio Lottery and the Megabucks Doubler from Massachusetts Lottery are both 6/49 games. The table above shows that the supposed odds for winning the jackpot in Ohio Lottery Classic Lotto are 1 in 13,983,816.

An observant reader will immediately question whether or not the information is valid. Either the title for the column is incorrect or the respective entries for odds are inaccurate.

It is important that you establish an accurate interpretation of data based on your knowledge about odds and probability. Do not accept what you read as it is.

Don’t you think that perhaps the column should be named “Probability” instead of “Odds”? Let me explain.

A 6/49 game has a total possible combination of 13,983,816. Therefore, if it is really the odds, it should have contained 1 to 13,983,815 instead of 1 in 13,983,816.

This 1 in 13,983,816 is a more appropriate as the probability to win, instead of odds.

Let me show you other examples of confusing odds tables.

More perplexing odds tables

The Virginia Lottery Cash 5 is a 5/41 game. The total possible combination in this game is 749,398.

Applying what we learned about probability and the formula above, the probability to win Cash 5 is 1 in 749,398.

Using the formula above for odds, we could get 1 to 749,397 as the odds to win in Cash 5.

Thus, do not feel confused when you visit the web page for Virginia Lottery Cash 5. You know better than to immediately believe that the odds of winning the jackpot are 1 in 749,398.

Our next figure is for California Lottery Fantasy 5. A 5/39 game like this has the total possible combinations of 575,757.

If we do the simple computation, we could get

Probability

= favorable combination / total possible combinations

= 1 / 575,757

Odds

= favorable combination/ (total possible combinations – favorable combinations)

= 1 / (575,757 – 1)

= 1/ 575,756

Thus, what interpretation can you give for the odds information in the table above? Is 1 in 575,757 probability or odds?

Next, we look at the of Lotto 6/42 from Louisiana Lottery.

It claims that the odds to win the cash jackpot in Louisiana Lottery Lotto are 1 in 5,245,786.

A 6/42 like this has the total possible combinations of 5,245,786.

Let me show you the simple math computations for probability and odds.

Probability

= favorable combination / total possible combinations

= 1 / 5,245,786

Odds

= favorable combination/ (total possible combinations – favorable combinations)

= 1 / (5,245,786- 1)

= 1/ 5,245,785

Therefore, the 1 in 5,245,785 from the table above is not the odds, but the probability.

Let as look now at this table for Hoosier Lottery Lotto 6/46 and see if the information is correct.

In a 6/46 game, the total number of possible combinations is 9,366,819.

Probability

= favorable combination / total possible combinations

= 1 / 9,366,819

Odds= favorable combination/ (total possible combinations – favorable combinations)= 1 / (9,366,819- 1)= 1/ 9,366,818

Would you believe what the table says that the odds to win the jackpot are 1 in 9,366,819?

It really helps to first confirm if the information you read is correct or not.

Our next example of confusing odds table is from Minnesota Lottery Northstar Cash. This is a 5/31 game that has 169,911 total possible combinations.

Let us see if the information of odds from the table is acceptable.

Probability

= favorable combination / total possible combinations

= 1 / 169,911

Odds= favorable combination/ (total possible combinations – favorable combinations)= 1 / (169,911 – 1)= 1/ 169,910

Do you just accept that the odds of winning the jackpot for Northstar Cash are 1 in 169,911?

A 6/47 game like the Classic Lotto 47 from Michigan Lottery has the total possible combinations of 10,737,573.

Looking at the values underneath the Odds column of the table above could make you get more confused. Sure, the title of the column is Odds. The succeeding entries even follow the depiction x: y that we mentioned above as applicable for odds.

Yet, are the numerical values acceptable?

Probability

= favorable combination / total possible combinations

= 1 / 10,737,573

Odds= favorable combination/ (total possible combinations – favorable combinations)= 1 / (10,737,573- 1)= 1/ 10,737,572

Our computations show that 1: 10,737,573 are not the odds for winning the jackpot in Classic Lotto 47. It is also not even the probability for the same game.

A similar game is this Jumbo Bucks Lotto from Georgia Lottery. See the image below.

Although different in the way of writing the figures, the values in this table from Georgia Lottery also do not conform to the values we have gathered from our odds computation.

It is more appropriate to say that 1: 10,737,573 is the probability to win the jackpot rather than the odds.

There are 45,057,474 total possible combinations in a 6/59 game like New York Lotto.

From the image shown from New York Lottery for the said game, the odds of winning the jackpot are 1 in 45,057,474.

Let us confirm if this is really the odds.

Probability

= favorable combination / total possible combinations

= 1 / 45,057,474

Odds= favorable combination/ (total possible combinations – favorable combinations)= 1 / (45,057,474- 1)= 1/ 45,057,473

From our computation, we found out that 1 in 45,057,474 refers to the probability instead of the odds.

Are you still not convinced that you must look closely at the information you read from some state lotteries?

Look at this example for Texas Lottery Lotto Texas whose total possible combinations are 25,827,165. We once again see that the title for the column is Odds, and the values under it follow the accepted configuration for odds.

We can compute for the odds and probability.

Probability

= favorable combination / total possible combinations

= 1 / 25,827,165

Odds

= favorable combination/ (total possible combinations – favorable combinations)

= 1 / (25,827,165- 1)

= 1/ 25,827,164

Therefore, 1: 25,827,165 does not refer to the odds or to the probability.

This coincides with the observation of Lawrence Fulton, Francis A. Méndez Mediavilla, Nathaniel Bastian, and Rasim Muzaffer Musal who prepared the manuscript entitled “Confusion Between Odds and Probability, a Pandemic?”

This manuscript appeared in the November 2012 edition of the Journal of Statistics Education. Let me quote the exact statement, which is

The figure presented as odds is indeed not odds.Source: Confusion Between Odds and Probability, a Pandemic?

This document talks about the common confusion from using the words odds and probability. The goal of this work coincides with the aim of this article. It points out this issue of emphasizing the importance and the responsibility of meticulously disseminating information.

It further supports the results of our computation and analysis that the value for odds to win the jackpot in Lotto Texas is not really the odds. The document also presented proof that the Texas Powerball incorrectly calculated odds and incorrectly reported probability as odds.

Our examples here are only a few. You could see a lot of misinterpretation and misrepresentation across many state lottery websites. Thus, do not be a victim of this confusion pandemic. Always remember, odds and probability are not mathematically equivalent.

Consider this as a reminder that you must double-check the information about odds and probability. After all, you need to make the best decision out of accurate data from mathematical analysis.

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