Math

QuestionDraw a card from a deck: win \11forafacecard,$9foranace,lose$6otherwise.Findtheexpectedgain:11 for a face card, \$9 for an ace, lose \$6 otherwise. Find the expected gain: -3.87$.

Studdy Solution

STEP 1

Assumptions1. The deck is a standard deck of52 cards. . There are4 aces,12 face cards (jacks, queens, and kings), and the remaining36 cards are neither aces nor face cards.
3. The winnings are 11forafacecard,11 for a face card, 9 for an ace, and a loss of $6 for any other card.
4. We are calculating the expected gain from the player's point of view.

STEP 2

First, we need to calculate the probabilities of drawing an ace, a face card, and any other card.The probability of an event is calculated as the number of ways the event can occur divided by the total number of outcomes.(Ace)=NumberofacesTotalnumberofcards(Ace) = \frac{Number\, of\, aces}{Total\, number\, of\, cards}(Facecard)=NumberoffacecardsTotalnumberofcards(Face\, card) = \frac{Number\, of\, face\, cards}{Total\, number\, of\, cards}(Othercard)=NumberofothercardsTotalnumberofcards(Other\, card) = \frac{Number\, of\, other\, cards}{Total\, number\, of\, cards}

STEP 3

Now, plug in the given values for the number of aces, face cards, other cards, and total cards to calculate the probabilities.
(Ace)=52(Ace) = \frac{}{52}(Facecard)=1252(Face\, card) = \frac{12}{52}(Othercard)=3652(Other\, card) = \frac{36}{52}

STEP 4

Calculate the probabilities.
(Ace)=452=0.07692307692(Ace) = \frac{4}{52} =0.07692307692(Facecard)=1252=0.23076923077(Face\, card) = \frac{12}{52} =0.23076923077(Othercard)=3652=0.69230769231(Other\, card) = \frac{36}{52} =0.69230769231

STEP 5

Now that we have the probabilities, we can calculate the expected gain. The expected value of a random variable is the sum of the products of each outcome and its probability.
(X)=x1(x1)+x2(x2)+x3(x3)(X) = x1(x1) + x2(x2) + x3(x3)where x1x1, x2x2, and x3x3 are the outcomes (winnings or losses) and (x1)(x1), (x2)(x2), and (x3)(x3) are their respective probabilities.

STEP 6

Plug in the values for the outcomes and their probabilities to calculate the expected gain.
(X)=$11×0.23076923077+$9×0.07692307692$6×0.69230769231(X) = \$11 \times0.23076923077 + \$9 \times0.07692307692 - \$6 \times0.69230769231

STEP 7

Calculate the expected gain.
(X)=$11×0.23076923077+$9×0.07692307692$6×0.69230769231=$3.87(X) = \$11 \times0.23076923077 + \$9 \times0.07692307692 - \$6 \times0.69230769231 = -\$3.87The expected gain from this game is -$3.87.

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