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4 Spot Keno
For the 4 Spot ticket the number of ways to catch all four is combin(20,4) = 4845. The number of ways to catch 3 and miss one is the product of the number of ways to choose 3 out of 20 and the number of ways to choose 1 out of 60, or combin(20,3) * 60 = 1140*60 = 68,400. The number of ways to catch 2 and miss 2 is the product of the number of watch to choose 2 out of 20 winners and 2 out of 60 losers, or combin(20,2)*combin(60,2) = 190 * 1770 = 336,300.
The return of the 4 Spot ticket is 1,122,900/1,581,580 = 71.00%.
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4 Spot Special
Next, let’s look at the 4 Spot Special. We just worked out all the combinations, so we only need to apply them to a different pay table as follows, based on a $7 ticket.
Catch
| Formula
| Combinations
| Pays
| Return
| 4
| combine(20,4)*combine(60,0)
| 4845
| $1360
| $6589200
| 3
| combine(20,3)*combine(60,1)
| 68400
| $20
| $1368000
| 2
| combine(20,2)*combine(60,2)
| 336300
| $0
| $0
| 1
| combine(20,1)*combine(60,3)
| 684400
| $0
| $0
| 0
| combine(20,0)*combine(60,4)
| 487635
| $0
| $0
| Total
|
| 1581580
| $0
| $7957200
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The expected return of the ticket is 7,957,200/1,581,580 = $5.0312. However we must divide this by $7, the cost of the ticket, to get the expected return, which is $5.0312/$7 = 71.87%. So, if you don’t mind extra volatility, and were going to bet $7 anyway, the 4 Spot Special is a better bet than the regular 4 Spot.
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