Gambling and Expected Value: Encore (OLG)

In this post on Gambling and Expected Value, we look at the "Encore" lottery offered by the OLG.
Click here to find similar posts on other lotteries and games of chance.

Encore (OLG)

Encore is a lottery offered by the Ontario Lottery and Gaming Corporation (OLG). 

How the Game Works

Encore is an add-on lottery you can play if you are already playing another lottery (Lotto 649, Lotto Max, Lottario, Ontario 49, Pick 2, Pick 3, Pick 4, or Daily Keno).The player chooses how many wagers he or she wishes to make (you can choose to play 1 to 10 combinations per ticket). The player doesn't get to choose their numbers though; a random string of seven digits (from 0 to 9) is chosen for him for each play. Winnings are determined by matching particular digits in the sequence.

Probabilities and Prizes

There are 22 different ways to win a prize playing Encore, ranging from $2 to $1,000,000 on a $1 wager. Figuring out their respective probabilities isn't too difficult, but does require some care to avoid overcounting. I've made the table below to help you see how to calculate how many possible sequences of numbers could win a particular prize.

Since the sequence consists of 7 randomly selected digits that can be repeated, there are a total of 10,000,000 (10 raised to the 7th power) possible number combinations that can be played. The probability of winning a particular prize is the number of ways to get a match divided by the total number of possible plays. For instance, the probability of winning the prize for matching only the last 4 digits is 891 divided by 10,000,000 (i.e. 0.00891%, or approximately 1 in 11,223). The table below shows the payout and the likelihood of winning for each of the 22 ways a player can win at Encore.

Subtracting the probabilities of the 22 ways to win from 100% gives the probability of losing at Encore. The probability of losing works out to exactly 891 in 1,000, or 89.1%.

Something you should note looking at the table of payouts and probabilities is that the prize doesn't consistently scale to the probability of occurrence. For instance, matching the First 2 and Last 4 digits has the same probability of matching the Last 6 digits, but the former pays only about 0.1% as much as the latter.

Example

The following four combinations were played in an Encore draw where the winning numbers 5554321. 

'A' = 5556789

'B' = 5556781

'C' = 7777321

'D' = 4321555 

'A' wins a $10 payout for matching only the first 3 digits. 'B' wins a $12 payout matching the first 3 and the last 1 digits. 'C' wins a $10 payout for matching only the last 3 digits. 'D' loses.

Expected Value

The expected value of is the sum of the products of the probability of each outcome and the monetary gain of that outcome. The net monetary gain is the payout less the wager. Encore has 23 possible outcomes (22 ways to win and 1 way to lose), so the expected value of Encore is the sum of 23 terms. The expected value  of Encore is:

Expected value calculation for the OLG lottery "Encore".

The expected value of Encore is -$0.486 per $1 wagered. This means that, on average, for every dollar you spend playing Encore, you lose 48.6 cents. As always, it's smarter not to gamble at all, though an expected value better than -$0.50 per $1 wager is slightly better than most lotteries. Therefore, if you are going to buy lotteries anyway, Encore appears to be one of the better choices.

From our analysis we can draw the following conclusions:

1. Encore has an expected value of slightly better than -$0.50 per $1 wager, which is comparable to other lotteries and typical 50/50 draws.
2. Accordingly, Encore is similar to other lotteries in that it you will generally lose money rapidly if you play.
3. Encore is neither the best choice, nor the worst choice, as far as OLG lotteries go.
4. Most of Encore's 22 ways to win have a very low probability of occurrence.
5. Encore's "22 ways to win" is a clever marketing trick to disguise the fact that there's an 89.1% chance you'll lose playing Encore. 
6. Encore's prizes do not scale with their probability of occurrence. Most of the prizes are much smaller than one should expect given their probability of occurrence.