Find the expected value of a $1 bet in six-spot keno if three winning spots pays $1 (but you pay $1 to play, so you actually break even), four winning spots pays $3 (but you pay $1 to play, so you profit $2), five pays $100, and six pays $2,600.
To find the expected value of a $1 bet in six-spot keno, we need to calculate the weighted average of the possible outcomes.
Let's calculate the expected value step-by-step:
1. Calculate the probability of winning in each scenario:
- Three winning spots: There are a total of 6C3 = 20 ways to choose 3 winning spots out of 6. Each selection has a probability of (3/80) * (77/79) * (78/78) = 0.05797.
- Four winning spots: There are a total of 6C4 = 15 ways to choose 4 winning spots out of 6. Each selection has a probability of (3/80) * (2/79) * (77/78) * (78/78) = 0.00381.
- Five winning spots: There are a total of 6C5 = 6 ways to choose 5 winning spots out of 6. Each selection has a probability of (3/80) * (2/79) * (1/78) * (77/78) * (78/78) = 0.00006.
- Six winning spots: There is only 1 way to choose all 6 winning spots out of 6, with a probability of (3/80) * (2/79) * (1/78) * (77/78) * (76/77) * (78/78) = 0.00000154.
2. Calculate the expected value:
Expected value = (Probability x Outcome) + (Probability x Outcome) + (Probability x Outcome) + (Probability x Outcome)
= (0.05797 x $0) + (0.00381 x $2) + (0.00006 x $100) + (0.00000154 x $2,600)
= $0 + $0.00762 + $0.006 + $0.004
Expected value = $0.01762
Therefore, the expected value of a $1 bet in six-spot keno is $0.01762.
To find the expected value of a $1 bet in six-spot keno, we need to multiply the payoff for each outcome by its corresponding probability and sum them up.
First, let's calculate the probability of each outcome:
- The probability of hitting exactly three winning spots out of six can be calculated using the combination formula: C(6, 3) = 20. The probability of this happening is 20 / 6^6, because there are 6^6 possible combinations of numbers when choosing 6 out of 80.
- The probability of hitting exactly four winning spots out of six can be calculated using the combination formula: C(6, 4) = 15. The probability of this happening is 15 / 6^6.
- The probability of hitting exactly five winning spots out of six can be calculated using the combination formula: C(6, 5) = 6. The probability of this happening is 6 / 6^6.
- The probability of hitting all six winning spots out of six can be calculated using the combination formula: C(6, 6) = 1. The probability of this happening is 1 / 6^6.
Now, let's calculate the expected value:
- When you hit exactly three winning spots, you break even, meaning your expected payoff is $0.
- When you hit exactly four winning spots, you profit $2 ($3 - $1), and the probability of this happening is 15 / 6^6.
- When you hit exactly five winning spots, you win $100, and the probability of this happening is 6 / 6^6.
- When you hit all six winning spots, you win $2,600, and the probability of this happening is 1 / 6^6.
To calculate the expected value, we multiply each payoff by its corresponding probability:
(0 * (20 / 6^6)) + (2 * (15 / 6^6)) + (100 * (6 / 6^6)) + (2600 * (1 / 6^6))
Simplifying this expression gives us the expected value of a $1 bet in six-spot keno.