Pick a Card.
Mike and dave play the following card game. Mike picks a card from the deck
If he selects a heart Dave gives him $5, if not, he gives Dave $2.
Determine Mikes's expectation.
Determine Daves' expectation
Can you help me with this? i have tried to do it and i came up with (Mike's = -$1.58 and dave's = - $4.79) but I don't think the game is "fair" so I don't think the expected value is 0. I am really confused!
Prob(heart) = 1/4
prob(not a heart) = 3/4
Mike's expectation:
he will receive $5 if he picks a heart
expectation = 5(1/4) = $1.25
Dave's expectation:
he will receive $2 if Mike does not pick heart
expectation = $2(3/4) = $1.50
The game is clearly in Dave's favour.
If you wanted to know what Dave should receive to make it a "fair" game, let the amount Dave should receive be $x
x(3/4) = 5(1/4)
x = $1.67
(how ever did you get -$1.58 and -$4.79 ???????)
To determine Mike's expectation and Dave's expectation in this card game, we need to calculate the probability of Mike selecting a heart and the probability of him not selecting a heart.
There are 52 cards in a standard deck, and 13 of them are hearts. So, the probability of Mike selecting a heart is 13/52 or 1/4. And the probability of him not selecting a heart is 1 - 1/4 = 3/4.
Now, let's calculate Mike's expectation:
- If Mike selects a heart, Dave gives him $5. So, the expected value in this case is 1/4 * $5 = $1.25.
- If Mike doesn't select a heart, Dave gives him -$2. So, the expected value in this case is 3/4 * (-$2) = -$1.50.
To find Mike's overall expectation, we need to sum up the expected values for each possibility:
Overall Expectation = (Probability of Selecting a Heart * Expected Value when Selecting a Heart) +
(Probability of Not Selecting a Heart * Expected Value when Not Selecting a Heart)
Therefore, Mike's expectation is:
Overall Expectation = (1/4 * $1.25) + (3/4 * -$1.50) = $0.3125 - $1.125 = -$0.8125
So, Mike's expectation is -$0.8125.
Now, let's calculate Dave's expectation:
Dave's expectation will be the negative of Mike's expectation since if Mike gains money, Dave loses money, and vice versa.
Therefore, Dave's expectation is -$(-$0.8125) = $0.8125.
Hence, Mike's expectation is -$0.8125, and Dave's expectation is $0.8125.
It is important to note that a fair game would have an expected value of 0 for both players. In this case, since Mike's expectation is negative and Dave's expectation is positive, the game is not fair as it favors Dave.