## Wednesday, 6 August 2014

### Gambling and Expected Value: Extra (WCLC)

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

### Extra (WCLC)

Extra is a lottery offered by the Western Canada Lottery Corporation (WCLC).

### How the Game Works

Extra is an add-on lottery you can play if you are already playing another lottery (Lotto 649, Lotto Max, Western Max, Western 49, or Pick 3). The player chooses how many wagers he or she wishes to make. Players are limited to a maximum of one wager on Extra per wager on the other lotteries. For example, if you play three sets of numbers on Lotto 649, you can choose to enter the Extra draw up to 3 times. The player doesn't get to choose their numbers in Extra; a random string of seven digits is chosen for him for each play. The first digit is always between 1 and 6, inclusive. The remaining digits can be from 0 to 9. Winnings are determined by matching particular digits in the sequence.

### Probabilities and Prizes

There are 7 different ways to win a prize playing Extra, ranging from \$2 to \$250,000 on a \$1 wager. Since the sequence consists of 7 randomly selected digits that can be repeated and the first of which is limited to between 1 and 6, there are a total of 6,000,000 (= 6*10*10*10*10*10*10) 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 the last 4 digits (but not the last 5, last 6, or all 7) is 540 divided by 6,000,000 (i.e. 0.009%, or approximately 1 in 11,111). The table below summarizes the prize payout and the probability of occurrence for all the ways you can win playing Extra.

The probability of losing at Extra (equivalently, the probability of not winning at Extra) is 100% minus the sum of all the probabilities of the winning outcomes. That works out to a 90% chance of losing.

It is worth noting that the prizes for the Last 5 and Last 4 digits are much smaller than the other prizes relative to the probability of winning. The probability of matching the Last 6 numbers is approximately 11 times smaller than the probability of matching the Last 5 numbers, but the prize is 100 times larger.

### Example

The following four combinations were played in an Extra draw where the winning numbers 5554321.
'A' = 5556789
'B' = 5556781
'C' = 6664321
'D' = 4321555

'A' and 'D' lose. 'B' wins a \$2 payout matching the last digit. 'C' wins a \$100 payout for matching the last 4 digits.

### 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. Extra has 8 possible outcomes (7 ways to win and 1 way to lose), so the expected value of Extra is the sum of 8 terms:

 Expected value calculation for the WCLC lottery "Extra".

The expected value of Extra is -\$0.542 per \$1 wagered. This means that, on average, for every dollar you spend playing Extra, you lose 54.2 cents. That's roughly 12% worse than the expected value of the similar lottery, Encore, run by the OLG. It's also worse than the WCLC's Pick 3 lottery.

From our analysis we can draw the following conclusions:

1. Extra's expected value is a loss of 54.2 cents per dollar wagered.
2. If you are going to spend money on lotteries anyway, Extra is a comparatively poor choice. There are other lotteries offering better expected value.
3. The prizes for matching the Last 4 or Last 5 numbers are an order of magnitude smaller than expected if all the prizes were scaled according to the likelihood of a payout.

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