Gambling and Expected Value: Pick 2 and Pick 4 (OLG)

In this post on Gambling and Expected Value, we look at two more "Pick" lotteries, Pick 2 and Pick 4.
Click here to find similar posts on other lotteries and games of chance.

Pick 2 (OLG)

Pick 2 is a lottery offered by the Ontario Lottery and Gaming Corporation (OLG). 

How the Game Works

The player chooses two numbers from '0' to '9'. Numbers can be repeated, so there are 100 possible selections. Players win if their selection is an exact match to the winning draw (i.e. match both numbers in the same order). A secondary prize is offered to players who match only the first digit (also in the same order). 

Probabilities and Prizes

The probability of matching both digits in the correct order is 1/100 (1%), with a prize of $99 on a $2 wager ($49.50 per $1 wagered). The probability of matching only the first digit is 9/100 (9%) and nets a prize of $2 on a $2 wager ($1 per $1 wagered). You might be wondering why it's a 9% chance of getting the smaller prize instead of 10%. There is a 1 in 10 chance (10%) of correctly matching the first number. However, that includes the probability of also matching the second number. We needed the probability of matching the first digit but not the second digit. In other words, the probability of matching the first number minus the probability of matching both: 10% - 1% = 9%. 

Note: This game costs $2 to play, but for analysis it's convenient to normalize prizes to dollars won per dollar wagered. It gives us a consistent basis to compare different lotteries or games that have varying minimum wagers.

Example

Four gamblers, 'A', 'B', 'C', and 'D' play Pick 2. 

'A' chooses 42.

'B' chooses 24.

'C' chooses 43.

'D' chooses 22. 

All four of them bet $2. The winning numbers drawn are 42. 

'A' wins $99. 

'C' wins $2. 

'B' and 'D' lose.

Expected Value

The expected value of Pick 2 is the weighted average value of the monetary gains or losses incurred from playing. That's pretty straightforward in Pick 2 because there are only three possible outcomes with only one way to play the game. Multiply the net gain from each outcome by the probability of that outcome, then sum for all outcomes. Keep in mind that the cost of playing must be subtracted from the winnings in order to calculate net gain. The expected value, expressed in dollars per dollar wagered is:

(48.50)×(1/100) + (0)×(9/100) + (-1)×(9/10) = -0.415

This means that, on average, you lose 41.5 cents for every dollar wagered playing Pick 2. Better for your bank account than playing the WCLC's version of Pick 3, analyzed previously, but still a fairly efficient way throw your money away.

Pick 4 (OLG)

Pick 4 is also a lottery offered by the OLG. 

How the Game Works

Similar to Pick 2, the player chooses numbers from '0' to '9' (four choices this time instead of two). Numbers can be repeated, so there are 10,000 possible selections. There are two ways to play: Straight and Box. In Straight play, the order of the digits matters. The player must match the winning numbers in the same order in order to win. 

In Box play, the order of the digits doesn't matter, just their values. 1334 is a match to 3341 in Box play, but not in Straight play. The benefit is that you increase your probability of winning in Box play, but the drawback is that the prize is reduced accordingly. Within Box play there are four possibilities, depending on the player's digit selections: 4-way, 6-way, 12-way, and 24-way. 4-way means there are four unique ways to arrange the four digits, which means there are only two unique digits (one of them is repeated twice; e.g. 1222). Likewise, 6-way, 12-way, and 24-way mean 6, 12, and 24 unique arrangements are possible. 6-way means there are two sets of digits that are the same (e.g. 1212). 12-way happens with three unique digits (one is repeated; e.g. 1231) and 24-way happens with four unique digits (e.g. 1234). 

Probabilities and Prizes

The various prizes available in Pick 4, and their respective probabilities, are summarized in the table below.

Expected Value

So there are five ways to play Pick 4, but because of some rounding by the OLG, some choices are slightly worse than others. For Straight play, the expected value is:

(4999)×(1/10000) + (-1)×(9999/10000) = -0.50

But for 24-way Box play, the expected value is:

(199)×(0.0024) + (-1)×(0.9976) = -0.52

The table below shows the expected value for all five possible ways to play Pick 4.

From the expected value we can draw the following conclusions:

1. If you're going to play the lottery anyway, Pick 2 is a smarter choice (you'll lose money quicker in the long run playing Pick 4).
2. If you're going to play Pick 4, it's smarter to choose the Straight play option or the Box option with a 4-way combination.