The math behind Lotto 47

Michigan Lottery’s Lotto 47 is an attractive lottery game with a rollover jackpot and fixed prizes. It also offers additional draws through the game’s add-ons.

One of the add-ons, EZMatch, offers instant random prizes at the moment of ticket purchase. Anytime the jackpot is won, it is set back to $1 million for the next draw.

Category

Lotto 47 is a combinatorial lottery jackpot game drawing from one pool: six numbers drawn from 1 – 47 (the base draw).

• Size of the draw: 6
• Size of a played line: 6
• Cost of a line: $1

The game has two add-ons: Double Play, allowing the same ticket to participate in a second similar draw (additional cost $1), and EZMatch, based on an instant number draw at the ticket purchase and a prize draw, assigning random prizes to said numbers (additional cost $1).

Prize schedule

There are four tiers of prizes in the base draw, defined as matching 3 to 6 numbers. They are all fixed prizes, except the top prize (6 numbers matched), which is a rollover jackpot. In the Double Play option, the four tiers remain the same, the fixed prizes are doubled, and the top prize is fixed at $1.5 million.

The prizes do not increase uniformly from the lowest to the highest. There are abrupt jumps between consecutive tiers, including for the corresponding odds of winning.

There are 11 possible fixed prizes assigned to EZmatch, from $2 to $500.

The odds of winning

The odds of winning in the base draw with one line range from 1 in 10,737,573 (for the jackpot) to 1 in 50 (for the lowest prize). The overall odds of winning a prize in the base draw are 1 in 47.

The same odds stand for the Double Play draw. If Double Play is activated, the odds of winning in at least one of the two draws are noted in the next table:

The odds of matching none to six numbers in the EZmatch option are noted in the next table:

The odds for any match in EZmatch are 1 in 1.72.

The odds of matching at least one number in EZmatch or matching at least 3 numbers in the base draw or both events are 1 in 1.695 or 58.99% probability.

If you match exactly one number in the EZmatch, the probability of winning at least $3 in EZmatch is 49.68%. If more than one number is matched, the prize exceeds $3 anyway (100% probability).

Average win and fairness

The next table lists the average win for each prize in the base draw:

For an average jackpot of $4 million*, the average win is $0.37.

Example of use and interpretation of the table: Over the long run, a win in the 5 of 6 match tier is distributed as an average win of $0.0573 per line played (assuming the prize is won with a relative frequency of about 1 in 43,649 draws).

The next table lists the fairness of each prize in the base draw relative to its probability:

For an average jackpot of $4 million*, its fairness would be 37.25%. However, this only applies in the circumstance where such a jackpot is actually won, as rollovers actually mean zero prizes in this tier.

*Note: The average jackpot was chosen as the average of the jackpots won over the last 10 years.

Expected value and house edge

The exact expected value of the Lotto 47 game cannot be determined because the first-tier prize is variable. We can still estimate it in two cases, associated with two assumptions:

a) The first case is taking the jackpot amount as zero for the base draw (case justified by a long series of draws when the jackpot is not won, given its probability).

b) The second case is taking an average amount for the jackpot won over a period of time (say, the last 10 years).

In case a), the expected value is –$0.728, or –72.89% as a percentage of the line cost. The house edge of the game is 72.89%.

In case b), the expected value is –$0.356, or –35.64% as a percentage of the line cost. The house edge of the game is 35.64%.

One way to interpret this is that in case a) you are expected to lose on average $0.728 at every $1 invested, over the long run.

The above values correspond to the base play.

The next calculations were made under the assumption that the published EZmatch odds are combination-level probabilities, not independent per-number prize probabilities. If the latter assumption were true, it would yield a positive expectation (negative house edge) in two of the situations below and the overall game.

With the Double Play option and no EZmatch option activated:

In case a), the expected value is –$1.318, or –65.90% as a percentage of the line cost. The house edge of the game is 65.90%.

In case b), the expected value is –$0.946, or –47.28% as a percentage of the line cost. The house edge of the game is 47.28%.

One way to interpret this is that in case a) you are expected to lose on average $1.318 at every $2 invested, over the long run.

With the EZmatch option and no Double Play activated:

In case a), the expected value is –$1.119, or –55.93% as a percentage of the line cost. The house edge of the game is 55.93%.

In case b), the expected value is –$0.746, or –37.31% as a percentage of the line cost. The house edge of the game is 37.31%.

One way to interpret this is that in case a) you are expected to lose on average $1.119 at every $2 invested, over the long run.

With both EZmatch and Double Play activated:

In case a), the expected value is –$1.708, or –56.93% as a percentage of the line cost. The house edge of the game is 56.93%.

In case b), the expected value is –$1.335, or –44.51% as a percentage of the line cost. The house edge of the game is 44.51%.

One way of interpreting this is that in case a) you are expected to lose on average $1.708 at every $3 invested, over the long run.

Final remarks

The low line cost makes the Michigan Lottery Lotto 47 suitable for syndicating and group play, especially with the Double Play option activated, which increases the odds of winning. However, this implies a high risk, as the second-tier prize is very low relative to its odds and compared to the jackpot.

As far as individual long-run play is concerned, the lowest house edge applies when the EZmatch option is activated.

In fact, the EZmatch option makes this game special, as matching one or two EZmatch numbers actually extends the base prize tier schedule from 3 – 6 to 1 – 6 numbers matched (even though in different instances), which is not common across combinatorial lotteries.

With a high probability of matching at least one number in EZ match (58.10%) and a decent probability of making any profit (19.90%) with EZmatch, not only does this option reduce the overall house edge of the game, but it also stands as a good premise to keep playing for regular small profits.