IntroductionOver 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
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.
- 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.
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.
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:
- 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
- luck or fortune: a game of chance
- possibility or probability of anything happening: a fifty-percent chance of success
"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
Example
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).
Always think in terms of Chances.
That way, you can't go wrong!
That way, you can't go wrong!
To learn more about the Odds / Chances discussions, you can read the following:
- Lottery Post Discussion Thread
- Odds and Probability - Brian Alspach
- Odds Slang for Probability? - Dr Math
- Lottery - Odds of winning (1 ticket and 2 tickets)- Ricci rec.gambling.lottery
- Lottery - Odds of winning (1 ticket and 2 tickets) - Perkis rec.gambling.lottery
- Lotteries: What are the odds?
Focus for June : Analysis of the Powerball Cash verses Annuity Options
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Stumble It!
Graph CS1 illustrates the minimum jackpots for each of the 7 Canadian lotteries. For Super 7, C$2.5 M is the fixed minimum, while C$3.0 M is the Lotto 649 minimum. Since both of the lotteries are nationwide, their jackpots will grow each time a drawing fails to produce a jackpot winner. Whereas, the provincial lottery jackpots are fixed at: C$1.0 M for the Atlantic 49, Ontario 49, and Western 649; and at C$2.0 M for BC 49 and Quebec 49.
Ticket prices for each of the lotteries also vary. The most expensive is Lotto 649, which costs C$2.0 per ticket. Next are BC 49 and Quebec 49, at C$1.0 per entry. While Super 7 sells 3 plays for C$2.00, the individual ticket costs C$0.67 each. Lastly, Atlantic 49, Ontario 49, and Western 649 tickets only cost C$0.50 each. Note however that Western 649 requires players to buy 2 games at once.
To make these purchases equivalent, let's assume that a player will purchase C$2.00 worth of tickets. For this amount: Lotto 649 players will have 1 chance to win the C$3.0M jackpot; the BC and Quebec 49 players will receive 2 chances to win C$2.0 M; Super 7 players will have 3 chances to win C$2.5 M; and the Atlantic 49, Ontario 49, and Western 649 players will have 4 chances to win C$1.0 M.
The overall odds of winning any prize also vary. Super 7 offers the best chances, giving players a 1:17.66 to win something. Super 7 odds are 1.8 times better than Lotto 469, BC 49 and Quebec 49 whose odds are 1:32.31. Further, Super 7 odds are 3.0 times better than the Atlantic, Ontario and Western 49 odds of 53.66.
Although Super 7 contains more winning tickets, the average ticket payout is C$4.92. Whereas, Lotto 649 tickets average the highest payout of C$20.54 (4.2 times better than Super 7). As for the provincials: Ontario 49 pays C$12.81 per ticket ( 2.6 times Super 7); Atlantic 49 pays C$14.97 (3.0 times); BC and Quebec 49 pay C$16.14 (3.3 times); and Western 649 pays out C$17.54 per ticket (3.6 times more than Super 7).
The total dollars returned to players is displayed as cents per dollar. These indicator is important because it allow us to compare value on an equivalent basis. As shown, Lotto 649 returns the least back to the players, at only C$0.32 per every dollar received. Next comes Super 7 at C$0.42 per dollar, and Ontario 49 at C$0.48/dollar. Both the BC and Quebec 49 give back 1/2 of their take. Atlantic 49 returns C$0.56 to players. And, Western 649 pays out a whopping C$0.65 per dollar received, mor than twice as much as Lotto 649.
