# Illinois Lotto with Extra Shot: Odds and Prizes

TR Smith is a product designer and former teacher who uses math in her work every day.

Lotto with Extra Shot, known simply as "Lotto" in Illinois, is a state lottery game with a jackpot prize and several smaller prize levels. The game is similar in structure to the national lottery games Powerball and Mega Millions, but the odds of winning are much better. Here is a complete analysis of how to calculate the probabilities of winning different prizes and the expected return.

## How to Play and Should You Buy Extra Shot?

Lotto tickets cost \$1 per board; on each board you select six distinct numbers from 1 to 52. You also have the option to pay an extra \$1 for a seventh number -- the Extra Shot -- that is used for a subset of the partial match prizes. The Extra Shot number is randomly drawn for you and ranges from 1 to 25; you don't get to choose it yourself. It may be equal to one of the regular six numbers. During the official lottery drawing, six numbers are selected for the base Lotto game, and a seventh number is drawn for players who've purchased the Extra Shot.

People who frequently play Illinois Lotto often wonder whether it is better to spend the extra \$1 for an Extra Shot number, or to buy a separate new lottery ticket instead. The answer is that it's better to spend the extra dollar on a separate board with six completely different numbers. This effectively doubles your chances of winning any prize, including the jackpot. In contrast, buying the Extra Shot option increases your changes of winning any prize by a much smaller margin, and does not increase your chances of winning the jackpot at all.

## Odds of Winning the Jackpot

Since Illinois Lotto is a 6-out-of-52 game, the probability of matching all six numbers and winning the jackpot is the reciprocal of (52 C 6), aka "52 choose 6." This is computed with the formula for combinations.

1 / [(52*51*50*49*48*47) / (6*5*4*3*2*1)]
= (6*5*4*3*2*1) / (52*51*50*49*48*47)
= 1 / 20358520
≈ 0.0000000491195
≈ odds of 1 in 20 million

The probability of winning the jackpot is 1 / 20358520 whether you buy Extra Shot or not. The jackpot starts at \$2 million and will grow as more people by tickets. If nobody has the winning ticket for a drawing, the prize amount rolls over to the next drawing. There is no prize for matching all six numbers and the Extra Shot.

## Odds of Matching 5/6, 4/6, 3/6 and 2/6 (Without Extra Shot)

Illinois Lotto awards smaller prizes for partial matches of the six drawn numbers. These prizes are not fixed amounts, but instead are awarded on a pari-mutuel basis, meaning the amount of the prize depends on the number of winners for each lower tier. The more winners there are, the smaller the prize.

The probability of matching exactly 5 out of 6 numbers chosen from 1 to 52 is equal to

[(46 C 1) * (6 C 5)] / 20358520

This is the number of ways to choose one wrong number times the number of ways to choose five correct numbers divided by the total number of ticket combinations. Simplifying gives you

46 * 6 / 20358520
= 69 / 5089630
≈ 0.000013557
≈ odds of 1 in 73762.75

The average prize amount for matching 5 out of 6 is about \$2,000. Here are the computed odds for the lower prizes of the base Illinois Lotto without an Extra Shot number.

• 4 out of 6: The probability is computed as
[(46 C 2) * (6 C 4)] / 20358520
= 1035 * 15 / 20358520
= 3105 / 4071704
≈ 0.00076258
≈ odds of 1 in 1311.34
The average prize amount for matching 4 out of 6 is about \$50.
• 3 out of 6: The probability is computed as
[(46 C 3) * (6 C 3)] / 20358520
= 15180 * 20 / 20358520
= 7590 / 508963
≈ 0.0149127
≈ odds of 1 in 67.06
The prize amount for matching 3 out of 6 is fixed at \$5.
• 2 out of 6: The probability is computed as
[(46 C 4) * (6 C 2)] / 20358520
= 163185 * 15 / 20358520
= 489555 / 4071704
≈ 0.120233
≈ odds of 1 in 8.32
The prize amount for matching 2 out of 6 is fixed at \$1.

## Odds of Partial Matches with Extra Shot

Since there are 25 possible extra shot numbers, the probability of matching X out of the 6 winning numbers plus the ES number is 1/25 times the probability of matching X out of 6 without ES. Illinois Lotto awards prizes for X = 5, 4, 3, 2, 1 and 0. There is no special prize for a perfect match of 6 out of 6 plus ES.

• 5 out of 6 + ES: The probability of winning is computed as
(69 / 5089630)(1 / 25)
= 69 / 127240750
≈ 0.000000542279
≈ odds of 1 in 1844068.84
The average prize for 5/6 + ES is about \$50,000.
• 4 out of 6 + ES: The probability of winning is computed as
= (1035 * 15 / 20358520)(1 / 25)
= 621 / 20358520
≈ 0.0000305032
≈ odds of 1 in 32783.45
The average prize for 4/6 + ES is about \$1,250.
• 3 out of 6 + ES: The probability of winning is computed as
= (15180 * 20 / 20358520)(1 / 25)
= 1518 / 2544815
≈ 0.000596507
≈ odds of 1 in 1676.43
The prize amount for 3/6 + ES is fixed at \$125.
• 2 out of 6 + ES: The probability of winning is computed as
= (163185 * 15 / 20358520)(1 / 25)
= 97911 / 20358520
≈ 0.00480934
≈ odds of 1 in 207.93
The prize amount for 2/6 + ES is fixed at \$25.
• 1 out of 6 + ES: The probability of winning is computed as
[(46 C 5)*(6 C 5)] / (20358520 * 25)
= 293733 / 18177250
≈ 0.0161594
≈ odds of 1 in 61.88
The prize amount for 1/6 + ES is fixed at \$10.
• 0 out of 6 + ES: The probability of winning is computed as
[(46 C 6)*(6 C 0)] / (20358520 * 25)
= 1338117 / 72709000
≈ 0.0184037
≈ odds of 1 in 54.34
The prize amount for 0/6 + ES is fixed at \$5.

## Overall Odds of Winning and Expected Return

The overall odds of winning any prize in the base lottery game are about 1 in 7.36. With the Extra Shot option, the odds of winning any prize are about 1 in 5.68. Mathematically, your odds of winning anything are better if you spend \$2 on two separate tickets with different numbers than to spend \$2 on a single ticket with Extra Shot.

In the base lottery game, the expected return per dollar spent is approximately \$0.3583. This is computed by multiplying the prize amounts by their respective probabilities, adding, and then dividing by the cost of a ticket.

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## Popular

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• Vic 22 months ago

I play infrequently but a coworker is trying to start a lottery pool in the office. He says he always buys the extra options on his tickets and I said it doesn't increase your chances of winning. I'm right, it doesn't!

So if he gets all 26 of us to chip in with \$1 he better come back with 26 regular tickets instead of only 13 tickets with worthless extras.