Studdy Solution
STEP 1
Assumptions1. A standard deck of cards has52 cards4 Aces,12 face cards (Jacks, Queens, Kings), and36 other cards (numbered through10).
. The winnings are \$12 for a face card, \$6 for an Ace, and a loss of \$18 for any other card.
3. The game is from the player's point of view.
STEP 2
First, we need to calculate the probability of drawing each type of card (face card, Ace, other card). The probability is the number of that type of card divided by the total number of cards.
(facecard)=TotalnumberofcardsNumberoffacecards(Ace)=TotalnumberofcardsNumberofAces(othercard)=TotalnumberofcardsNumberofothercards
STEP 3
Now, plug in the given values for the number of each type of card and the total number of cards to calculate the probabilities.
(facecard)=5212(Ace)=52(othercard)=5236
STEP 4
implify the probabilities.
(facecard)=5212=133(Ace)=524=131(othercard)=5236=139
STEP 5
Now, we can calculate the expected gain from the game. The expected gain is the sum of the product of the winnings for each type of card and the probability of drawing that type of card.
Expectedgain=∑(Winnings×Probability)
STEP 6
Plug in the values for the winnings and the probabilities to calculate the expected gain.
Expectedgain=($12×(facecard))+($6×(Ace))+(−$18×(othercard))
STEP 7
Substitute the calculated probabilities into the equation.
Expectedgain=($12×133)+($6×131)−($18×139)
STEP 8
Calculate the expected gain.
Expectedgain=($12×133)+($6×131)−($18×13)=−$.23The expected gain from this game is -\$.23. This means, on average, the player will lose \$.23 per game.