A game involves tossing two coins. If the result is two heads, you win 3 points. If the result is one tail and one head, you win 1 point. If the result is two tails, you lose 5 points. Is this a fair game?
Outcomes:
HH
HT
TH
TT
Prob(2 heads in 2 throws) = (1/2)(1/2) = 1/4
prob(1 h, 1t) = 2/4 = 1/2
prob(2 T) = 1/4
expected value = (1/4)(3) + (1/2)(1) - 5(1/4) = 0
yup, fair game, you can play it all night long
Well, let me put on my juggling shoes and calculate the fairness here! When life gives you two heads, you get 3 points. When it serves you up a nice mix of a tail and a head, you get 1 point. And when it decides to be a bit of a prankster and give you two tails, you lose 5 points.
Now, if we examine the probabilities, there are four possible outcomes: two heads, one tail and one head, one head and one tail (the order doesn't matter), and finally two tails. If we assign fair weights to these outcomes, the game will be fair.
So, let's grab our probability hats and check it out. The probability of getting two heads is 1/4, the probability of getting one tail and one head is also 1/4, and the probability of getting two tails is 1/4 as well.
Now, if we calculate the expected value, we have (1/4 * 3) + (1/4 * 1) + (1/4 * (-5)) = (3/4) - (5/4) = -1/2.
So, my friend, it seems like this game is slightly tilted in favor of the house. But hey, life isn't always fair, right?
To determine if the game is fair, we need to calculate the expected value.
The possible outcomes are:
- Two heads (HH): 3 points
- One tail and one head (TH or HT): 1 point
- Two tails (TT): -5 points
The probabilities of these outcomes are:
- P(HH) = Probability of getting two heads = (1/2) * (1/2) = 1/4
- P(TH or HT) = Probability of getting one tail and one head = 2 * (1/2) * (1/2) = 1/2
- P(TT) = Probability of getting two tails = (1/2) * (1/2) = 1/4
Now, let's calculate the expected value:
Expected value = (3 * P(HH)) + (1 * P(TH or HT)) + (-5 * P(TT))
= (3 * 1/4) + (1 * 1/2) + (-5 * 1/4)
= 3/4 + 1/2 - 5/4
= 3/4 - 5/4
= -2/4
= -1/2
Since the expected value is negative (-1/2), it means, on average, you lose half a point per game. Therefore, the game is not fair.
To determine if the game is fair or not, we need to calculate the expected value (or average value) of each possible outcome and see if it equals zero.
Let's calculate the expected value for each outcome:
1. Two heads: The probability of getting two heads is 1/4 (since there are four possible outcomes when tossing two coins: HH, HT, TH, TT, and only one of them results in two heads). Winning 3 points with a probability of 1/4 gives an expected value of (3 * 1/4) = 3/4.
2. One tail and one head: The probability of getting one tail and one head is 1/2 (since there are two possible outcomes: HT or TH, out of the four total outcomes). Winning 1 point with a probability of 1/2 gives an expected value of (1 * 1/2) = 1/2.
3. Two tails: The probability of getting two tails is 1/4 (similar to two heads). Losing 5 points with a probability of 1/4 gives an expected value of ( -5 * 1/4) = -5/4.
To find the overall expected value, we add up the expected values of each outcome:
(3/4) + (1/2) + (-5/4) = (6/8) + (4/8) + (-5/8) = 5/8
The overall expected value is 5/8, which means that on average, you win 5/8 of a point per game.
Since the expected value is positive (greater than zero), the game is considered favorable to the player.