To calculate the expected winnings, we need to multiply each possible outcome by its corresponding probability and then sum them up.
a) To find out what a player can expect to win by playing this game, we first need to find the probability of each outcome. Since the die has 6 sides, each number from 1 to 6 has an equal probability of 1/6.
Using the given table of winnings, we multiply each winning amount by its probability as follows:
Winning amount:
$10 * (1/6) = $10/6
$9 * (1/6) = $9/6
$8 * (1/6) = $8/6
$7 * (1/6) = $7/6
$6 * (1/6) = $6/6
$5 * (1/6) = $5/6
Now, let's calculate the expected winnings:
Expected winnings = (10/6) + (9/6) + (8/6) + (7/6) + (6/6) + (5/6)
Expected winnings = 45/6
Dividing numerator and denominator by 3 to simplify:
Expected winnings = 15/2
Therefore, a player can expect to win $15/2 or $7.50 by playing this game.
b) To find a fair value to pay to play this game, we need to calculate the expected winnings and subtract the cost of playing.
Fair value = expected winnings - cost of playing
Fair value = 15/2 - 7
Fair value = 15/2 - 14/2
Fair value = 1/2
Therefore, a fair value to pay to play this game would be $1/2 or 50 cents.