Keno Odds / Math
Probability is the likelihood or chance that something is the case or will happen.
Odds is the favour of an event or a proposition and is represented as the inverse of the probability.
Example 1: Suppose we have 5 apples and 1 of the 5 apples is a green apple. Then the probability of picking a green apple is 1/5 or expressed in odds as 5 to 1 against picking a green apple.
Combination is the quantity of subgroups of a size 'R' that can be formed out of a group of a size 'N' in which the order is NOT important. This expression is often written mathematically as C(N,R) where R is less than or equal to N.
Problem #1: Suppose 'N' is a group of the following elements:
N: {A,B,C,D,E} for a total of 5 elements
What is the quantity of subgroups of a size 'R'=2 in which order is NOT important, which is expressed mathematically as C(5,2)?
Solution #1: AB, AC, AD, AE, BC, BD, BE, CD, CE, DE - The answer is 10 subgroups.
Rather than manually determining all possible combinations, the following mathematical formula will yield the same results:
Combination of Subgroups = N! / R!(N-R)!, where ! is the mathematical symbol for factorial.
10 = 5! / 2!(5-2)!
Note: 5!=(5)(4)(3)(2)(1)=120 and 2!=(2)(1)=2 and 3!=(3)(2)(1)=6
Therefore, 120 divided by 12 equals 10
Problem #2: Given eleven marbles, how many subgroups of 5 marbles can be grouped together such that no subgroups contain the same 5 marbles, which is expressed mathematically as C(11,5)?
Solution #2: 11! / 5!(11-5)! = 462 subgroups.
Note: 11!=(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1)=39,916,800 and 5!=(5)(4)(3)(2)(1)=120 and 6!=(6)(5)(4)(3)(2)(1)=720
Therefore, 39,916,800 divided by 86,400 equals 462.
Problem #3: Suppose we select 5 numbers out of 80 numbers available on a Keno board, which a total of 20 random balls will be drawn from a bin. What is the probability and odds of getting 5 out of 5 ?
Solution #3: First, we must determine all possible Keno outcomes given the 20 random balls drawn from a bin, which 5 balls drawn are the same as the 5 numbers originally selected. Therefore, the remaining 15 balls would be a subset of the remaining 75 numbers on the Keno board. Since the arrangement of the 15 balls is not important, then all of the outcomes possible that contain the original 5 numbers selected is:
C(75,15)*C(5,5). Therefore, the probability of getting a 5 out of 5 is:
C(75,15)*C(5,5)/C(80,20)=0.000644924695559 and this is equivalent to an ODDS of 1/0.000644924695559=1550.56
This means there is a potential chance of hitting 5 out of 5 within 1,550.56 games.
Problem #4: Suppose we select 5 numbers out of 80 numbers available on a Keno board, which a total of 20 random balls will be drawn from a bin. What is the probability and odds of getting 3 out of 5 ?
Solution #4: First, we must determine all possible Keno outcomes given the 20 random balls, which 3 balls of the 20 random balls drawn are the same as the 5 numbers orginally selected. Therefore, the remaining 17 balls would be a subset of the remaining 75 numbers on the Keno board. Since the arrangement of the 17 balls is not important AND the arrangement of the 3 balls within the original 5 numbers selected is not important, all of the outcomes possible that contain any combination of 3 of the original 5 numbers selected is:
C(75,17)*C(5,3). Therefore, the probability of getting a 3 out of 5 is:
C(75,17)*C(5,3)/C(80,20)=0.839350522894 and this is equivalent to an ODDS of 1/0.839350522894=11.91
This means there is a potential chance of hitting 3 out of 5 within 11.91 games.
Combination Keno Pro calculates the expected return of a particular Keno spot played that is based upon the selected Keno Payout table being used. Assuming a $1 dollar payout table, the expected return is determined by summation of all probabilities of a win scenario for a "spot" being wagered times the payout of each hit scenario for the "spot". This is perhaps best demonstrated by an example using the 6 spot.
| 6 Spot - Bet $1.00 |
| Hits | Odds | Probability | Payout | Multiplication |
| 0 | 6.0 to 1 | 0.166667 | $0.00 | 0.00000 |
| 1 | 2.8 to 1 | 0.357143 | $0.00 | 0.00000 |
| 2 | 3.2 to 1 | 0.312510 | $0.00 | 0.00000 |
| 3 | 7.7 to 1 | 0.129870 | $1.00 | 0.129870 |
| 4 | 35.0 to 1 | 0.028571 | $3.00 | 0.085713 |
| 5 | 323.0 to 1 | 0.003096 | $90.00 | 0.278640 |
| 6 | 7,752.8 to 1 | 0.000129 | $1,500.00 | 0.193500 |
| Total Expected Return: |
0.687723 |
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This is equivalent to an expected return of approximately 0.69 cents for every dollar wagered or otherwise written at 68.77% percent expected return for this example.
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