Saturday, April 26, 2008

How Lottery Odds Change When Buying Multiple Tickets

Over the past several years, I have read many posts on the internet from people who have wondered how their odds of winning the large lottery jackpots improve when they purchase multiple tickets. Most often, the replies stated that you should simply:

Divide the
Number of tickets you purchased for that drawing.
by the
Total number of combinations

Is this correct?
The answer is: Absolutely yes. This is the basic formula for computing the probability.

For example:
If you play Powerball, the chances of your tickets matching the winning Jackpot combination when you buy:
  • 1 Ticket, will be: 1 in 146,107,962 or 1 in 146,107,962
  • 2 Tickets, will be: 2 in 146,107,962 or 1 in 73,053,981
  • 3 Tickets, will be: 3 in 146,107,962 or 1 in 48,702,654
  • 4 Tickets, will be: 4 in 146,107,962 or 1 in 36,526,990.5
  • 5 Tickets, will be: 5 in 146,107,962 or 1 in 29,221,592.4
  • and so on.
Similarily, if you play Mega Millions, the chances of your tickets matching the winning Jackpot combination when you buy:
  • 1 Ticket, will be: 1 in 175,711,536 or 1 in 175,711,536
  • 2 Tickets, will be: 2 in 175,711,536 or 1 in 87,855,768
  • 3 Tickets, will be: 3 in 175,711,536 or 1 in 58,570,512
  • 4 Tickets, will be: 4 in 175,711,536 or 1 in 43,927,884
  • 5 Tickets, will be: 5 in 175,711,536 or 1 in 35,142,307.2
  • and so on.
Note that because this is a simple division formula, it is mathematically correct to reduce the numerator and denominator to the lowest terms. Whether the terms are reduced or not, the resulting probability value will be identical.

If this is so simple, why do people argue about it?
Because of the definition of the word "Odds". If you visit the AllExperts.com post: Probability & Statistics, you will read that we are in agreement. However, you will note that the article speaks in terms of "Chances" and "Probability", but not "Odds".

The question by Daren Henning in Dr. Math's Powerball Odds When Buying More Tickets, also alludes to this confusion, but the answer is not clarified.

Mr. Henning says that he and his friend are arguing about the odds when buying 10 tickets in a hypothetical 80,000,000 Powerball lottery. The friend says the odds are 10/80,000,000 or 1 in 8,000,000. However, Henning thinks the odds are to 79,999,990 to 1. Dr. Math agrees with the friend and explains why.

To be correct, Henning should have stated that he believed the odds were 10 to 79,999,990. In this case, he too would be correct.

How can they both be correct?
Once again, because of the definition of the word "Odds"
It is context dependent.

Statistical Definitions
When you buy a Lottery Ticket, you are buying a "chance" to win the jackpot.

In Dictionary.com, chance is defined as:
  1. the absence of any cause of events that can be predicted, understood, or controlled: often personified or treated as a positive agency: Chance governs all
  2. luck or fortune: a game of chance
  3. possibility or probability of anything happening: a fifty-percent chance of success
Measuring Chance
"Chance is measured using either probabilities (a ratio of occurrence to the whole) or odds (a ratio of occurrence to nonoccurrence, or for and against)."
  • Probability = events/(events+non-events) values range from 0 to 1
  • Odds = events/non-events values range from 0 to infinity
From sources: Measuring chance and Primer on Probability, Odds and Interpreting their Ratios

As an example, assume that we are rolling a single 6 sided dice.

We wish to measure the chance that a "5" will appear. The probability that a 5 will appear is 1/6, or 0.1666667. Whereas, the odds that a 5 will appear are 1/5, meaning one change for, and 5 against.

Next, let us measure the chance that an even number will appear. The probability that an even number will appear is 3/6, or 0.50. Whereas, the odds that an even number will appear are 3/3 or 1/1 meaning one change for and one against.

Confusion Abounds
Just looking at the numbers above, one cannot tell if we are viewing an expression of probability or odds. While all these numbers are correct, we need more information in order to interpret their meanings properly.

Returning to Mr. Henning's question about the lottery odds above, both the friend and Dr. Math are asserting the correctness of the probability definition. In this, they are correct, but should have clearly stated that they are speaking of Probability, not Odds.

Whereas, according to the above definitions, Mr. Henning is correct in asserting the odds, because he is referring to the occurrences for verses occurrences against.

Resolving the Ambiguity of the Word Odds
We cannot change City Hall. In the Lottery World, the use of the word Odds is often synonymous with Probability or Chances.

If you visit the Minnesota Lottery Figuring the Odds or the Powerball - Prizes and Odds pages, you will see that they incorrectly define your odds in terms of Probability.

Conversely, when you read the Mega Millions How to Play and BCLC How play Lotto 6/49 pages, you will learn that they are correctly presenting your Chances.

In order to resolve the ambiguity of the word Odds, we believe the meaning is "sub-language" dependent.
  • Within the Lottery Context, we suggest that Odds be interpreted as Lottery Odds (which is the same as Probability)
  • Within the Gambling Context, such as horse racing, we suggest that Odds be interpreted as the defined Odds (Chances For verses Chances Against).
The easiest and least confusing thing to remember about Lottery Odds is to:

Always think in terms of Chances.
That way, you can't go wrong!

To learn more about the Odds / Chances discussions, you can read the following:

Focus for June : Analysis of the Powerball Cash verses Annuity Options

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  1. I agree with you about these. Well someday Ill create a blog to compete you! lolz.

  2. This is incorrect. Lookup binomial distribution or possion distibution.

  3. Max K,

    This article simply describes the differences between odds and chances, and correctly illustrates how your chances improve when you buy multiple combinations.

    We are quite familiar with binomial and Poisson distributions. These are statistical models that can be used to estimate the probability of various events occurring. If you search this blog, you will find 3 articles discussing Poisson. And, if you look at our hot cold number analysis pages, you will find that we utilize a binomial distribution to identify hotness and coldness.

    Unfortunately, when it comes to analyzing numbers, there are many measures that can be used. Each may be different, but they are all correct within their individual contexts.



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