The Tri-State Pick 3 lottery game offers a choice of several bets. You choosea three-digit number. The lottery commision announces the winning three-digit number, chosen at random, at the end of each day. The "box" pays $83.33 if the number you choose has the same digits as the winning number, in any order; Find the expected payoff for a $1 bet on the box. (Assume that you chose a number having three different digits.
I have tried this problem many times and cannot get the right answer.
To find the expected payoff for a $1 bet on the box in the Tri-State Pick 3 lottery game, you need to calculate the probability of winning and the amount won if you do win.
Let's break it down step by step:
1. Calculating the probability of winning:
In the Tri-State Pick 3 game, you choose a three-digit number, assuming all three digits are different. The winning three-digit number chosen at random has six possible combinations of digits (for example, if the winning number is 123, the combinations are 123, 132, 213, 231, 312, 321). There are a total of 10 x 9 x 8 = 720 possible three-digit numbers you could choose (10 options for the first digit, 9 options for the second digit, and 8 options for the third digit). Therefore, the probability of choosing the correct three-digit number is 6/720 or 1/120.
2. Calculating the amount won:
The box bet pays $83.33 if the number you choose has the same digits as the winning number, in any order.
3. Calculating the expected payoff:
The expected payoff is calculated by multiplying the probability of winning by the amount won and subtracting the cost of the bet.
Expected payoff = (Probability of winning) x (Amount won) - (Cost of bet)
In this case, the probability of winning is 1/120, the amount won is $83.33, and the cost of the bet is $1.
Expected payoff = (1/120) x ($83.33) - ($1)
Expected payoff = $0.694 - $1
Expected payoff ≈ -$0.306
Based on this calculation, the expected payoff for a $1 bet on the box in the Tri-State Pick 3 lottery game is approximately -$0.306. This means that, on average, you can expect to lose about $0.306 for each $1 bet you place on the box in the long run.
It's worth noting that this value represents the average outcome over a large number of bets. In any single instance, you could win or lose more or less than this expected value.