You and your friend each have a coin. For each round, each of you flips your coin. If both coins show heads, your friend pays you $3. If both show tails, your friend pays you $1. If the coins don't match, you pay him $2. Is this a fair game?
The four possible outcomes are:
outcome payment
H/H +3.00
H/T -2.00
T/H -2.00
T/T -1.00
Average -2.00/4 = -0.50
Decide for yourself.
mathmate~
in the last part of the list T/T you have a -1.00 but shouldnt it be +1.00? does that then change the final answer?
thank you!!
Definitely!
Sorry that I misread the payment scheme.
So it is now:
outcome payment
H/H +3.00
H/T -2.00
T/H -2.00
T/T +1.00
Average 0.00/4 = 0.0
I suppose the result is now evident.
thank you! :)
To determine if this is a fair game, we need to calculate the expected value of the game. The expected value represents the average amount of money you would win or lose per round if you played the game many times.
Let's consider the possible outcomes and their probabilities:
1) Both coins show heads:
Probability = 1/2 * 1/2 = 1/4
You win $3.
2) Both coins show tails:
Probability = 1/2 * 1/2 = 1/4
You win $1.
3) The coins don't match:
Probability = 1/2 * 1/2 = 1/4
You lose $2.
To calculate the expected value, we multiply the value for each outcome by its corresponding probability, and then sum them up:
Expected value = (1/4 * $3) + (1/4 * $1) + (1/4 * -$2)
Expected value = $3/4 + $1/4 - $2/4
Expected value = $2/4
Expected value = $0.50
The expected value of the game is $0.50. If the expected value is zero, it means the game is fair because over the long run, you would neither win nor lose money on average. In this case, the expected value is $0.50, so the game is not fair.
Therefore, this game is not fair, and over time you can expect to lose an average of $0.50 per round.