A game is played using three 6-sided dice. You throw these dice one time and if the dice all show the same number ( triples), you win $ 40.00 . For this game to be “fair”, how much should a person pay to play this game ?
well lets figure out probability of all ones, then multiply by 6
1/6 * 1/6 * 1/6 = 1/6^3
now probability of three of any old number is
6/6^3 = 1/36
40/36 = $1.11111 but long term the house should come out ahead :)
so
$1.12
To determine the fair price for this game, we need to calculate the probability of rolling triples and use that information to calculate the expected value.
First, let's find the probability of rolling triples. Since there are 6 possible outcomes on each dice (numbers 1 to 6), the probability of getting triplets is 1 out of 6^3 (six multiplied by itself three times) because there is only one combination on each dice that satisfies the condition. Therefore, the probability of rolling triples is 1/216.
Now, let's calculate the expected value. The expected value is the average outcome, taking into account the probabilities of all possible outcomes. In this case, there are two possible outcomes: winning $40 with a probability of 1/216 or losing the game with a probability of 1 - 1/216.
The expected value (EV) can be calculated using the following formula:
EV = (probability of winning * winning amount) + (probability of losing * losing amount)
Using this formula, the expected value for this game can be calculated as:
EV = (1/216 * $40) + ((1 - 1/216) * -$x)
Since the game is fair, the expected value should be zero, as the person should neither gain nor lose money on average. Therefore, we can set the equation to zero and solve for the unknown losing amount ($x).
0 = (1/216 * $40) + ((1 - 1/216) * -$x)
Now, we solve for $x:
0 = (1/216 * $40) - (1 - 1/216) * $x
(1 - 1/216) * $x = (1/216 * $40)
($215/216) * $x = $40/216
Next, we rearrange the equation to solve for $x:
$x = ($40/216) / ($215/216)
$x = $40 * (216/$215)
$x ≈ $40.00 * 1.004651
At this point, we find that $x is approximately $40.19. Therefore, a person should pay approximately $40.19 to play this game for it to be fair.