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QuestionDraw a card from a deck: win \$6 for a face card, \$5 for an ace, lose \$3 for others. Find the probability distribution and expected gain.

Studdy Solution

STEP 1

Assumptions1. The game is played with a standard deck of52 cards. . The deck contains4 aces,12 face cards (jacks, queens, and kings), and36 other cards (numbers through10 in four suits).
3. The winnings are 6forafacecard,6 for a face card, 5 for an ace, and a loss of $3 for any other card.
4. The probability of drawing a particular type of card is the number of that type of card divided by the total number of cards.

STEP 2

First, we need to calculate the probability of drawing a face card, an ace, and any other card. The probability is calculated as the number of that type of card divided by the total number of cards.
(face card)=Number of face cardsTotal number of cards(face\, card) = \frac{Number\, of\, face\, cards}{Total\, number\, of\, cards}(ace)=Number of acesTotal number of cards(ace) = \frac{Number\, of\, aces}{Total\, number\, of\, cards}(other card)=Number of other cardsTotal number of cards(other\, card) = \frac{Number\, of\, other\, cards}{Total\, number\, of\, cards}

STEP 3

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

STEP 4

implify the probabilities.
(face card)=1252=313(face\, card) = \frac{12}{52} = \frac{3}{13}(ace)=452=113(ace) = \frac{4}{52} = \frac{1}{13}(other card)=3652=913(other\, card) = \frac{36}{52} = \frac{9}{13}

STEP 5

Now, we can create the probability distribution table. The table has two columns xx for the winnings and (x)(x) for the probability of that winnings.
\begin{tabular}{|c|c|} \hlinexx & (x)(x) \\ \hline$$ & $\frac{3}{13}$ \\ $5$ & $\frac{1}{13}$ \\ $-3$ & $\frac{9}{13}$ \\ \hline\end{tabular}

STEP 6

To find the expected gain from this game, we need to multiply each possible winnings by its probability and then sum these products.
(x)=∑x⋅(x)(x) = \sum x \cdot(x)

STEP 7

Now, plug in the values from the probability distribution table to calculate the expected gain.
(x)=6⋅313+5⋅113−3⋅913(x) =6 \cdot \frac{3}{13} +5 \cdot \frac{1}{13} -3 \cdot \frac{9}{13}

STEP 8

Calculate the expected gain.
(x)=6⋅313+5⋅113−3⋅13=1813+513−2713=−413(x) =6 \cdot \frac{3}{13} +5 \cdot \frac{1}{13} -3 \cdot \frac{}{13} = \frac{18}{13} + \frac{5}{13} - \frac{27}{13} = -\frac{4}{13}The expected gain from this game is −413-\frac{4}{13} dollars. This means that, on average, the player will lose about 0.310.31 dollars each time they play the game.

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