You pick a card from a deck.If you get a club,you win $80.If not,you get to draw again (after replacing the first card) .If you get a club the second time,you win $30.Otherwise you win nothing. Create a probability model for the amount you win at this game.

A) $1421642764\begin{array} { l | r r r } \text { Amount won } & \$ 80 & \$ 30 & \$ 0 \\\hline \text { P(Amount won) } & \frac { 1 } { 4 } & \frac { 21 } { 64 } & \frac { 27 } { 64 }\end{array}$
B) $149643162764\begin{array} { l | c c c c c } \text { Amount won } & \$ 80 & \$ 60 & \$ 30 & \$ 0 \\\hline P ( \text { Amount won) } & \frac { 1 } { 4 } & \frac { 9 } { 64 } & \frac { 3 } { 16 } & \frac { 27 } { 64 }\end{array}$
C) $14316916\begin{array} { l | c c c } \text { Amount won } & \$ 80 & \$ 30 & \$ 0 \\\hline \mathrm { P } \text { (Amount won) } & \frac { 1 } { 4 } & \frac { 3 } { 16 } & \frac { 9 } { 16 }\end{array}$
D) $116316316916\begin{array} { l | c c c c c } \text { Amount won } & \$ 110 & \$ 80 & \$ 30 & \$ 0 \\\hline P ( \text { Amount won) } & \frac { 1 } { 16 } & \frac { 3 } { 16 } & \frac { 3 } { 16 } & \frac { 9 } { 16 }\end{array}$
E) $141412\begin{array} { l | r c c } \text { Amount won } & \$ 80 & \$ 30 & \$ 0 \\\hline \text { P(Amount won) } & \frac { 1 } { 4 } & \frac { 1 } { 4 } & \frac { 1 } { 2 }\end{array}$

Question

You pick a card from a deck.If you get a club,you win $80.If not,you get to draw again (after replacing the first card) .If you get a club the second time,you win $30.Otherwise you win nothing. Create a probability model for the amount you win at this game.A)     Amount won  $ 80 $ 30 $ 0  P(Amount won)   1 4 21 64 27 64 \begin{array} { l | r r r } 	ext { Amount won }  &  \$ 80  &  \$ 30  &  \$ 0 \\hline 	ext { P(Amount won)  }  &  \frac { 1 } { 4 }  &  \frac { 21 } { 64 }  &  \frac { 27 } { 64 }\end{array}  Amount won   P(Amount won)   ​ $80 4 1 ​ ​ $30 64 21 ​ ​ $0 64 27 ​ ​ ​ B)     Amount won  $ 80 $ 60 $ 30 $ 0 P (  Amount won)   1 4 9 64 3 16 27 64 \begin{array} { l | c c c c c } 	ext { Amount won }  &  \$ 80  &  \$ 60  &  \$ 30  &  \$ 0 \\hline P ( 	ext { Amount won)  }  &  \frac { 1 } { 4 }  &  \frac { 9 } { 64 }  &  \frac { 3 } { 16 }  &  \frac { 27 } { 64 }\end{array}  Amount won  P (  Amount won)   ​ $80 4 1 ​ ​ $60 64 9 ​ ​ $30 16 3 ​ ​ $0 64 27 ​ ​ ​ C)     Amount won  $ 80 $ 30 $ 0 P  (Amount won)   1 4 3 16 9 16 \begin{array} { l | c c c } 	ext { Amount won }  &  \$ 80  &  \$ 30  &  \$ 0 \\hline \mathrm { P } 	ext { (Amount won)  }  &  \frac { 1 } { 4 }  &  \frac { 3 } { 16 }  &  \frac { 9 } { 16 }\end{array}  Amount won  P  (Amount won)   ​ $80 4 1 ​ ​ $30 16 3 ​ ​ $0 16 9 ​ ​ ​ D)     Amount won  $ 110 $ 80 $ 30 $ 0 P (  Amount won)   1 16 3 16 3 16 9 16 \begin{array} { l | c c c c c } 	ext { Amount won }  &  \$ 110  &  \$ 80  &  \$ 30  &  \$ 0 \\hline P ( 	ext { Amount won)  }  &  \frac { 1 } { 16 }  &  \frac { 3 } { 16 }  &  \frac { 3 } { 16 }  &  \frac { 9 } { 16 }\end{array}  Amount won  P (  Amount won)   ​ $110 16 1 ​ ​ $80 16 3 ​ ​ $30 16 3 ​ ​ $0 16 9 ​ ​ ​ E)     Amount won  $ 80 $ 30 $ 0  P(Amount won)   1 4 1 4 1 2 \begin{array} { l | r c c } 	ext { Amount won }  &  \$ 80  &  \$ 30  &  \$ 0 \\hline 	ext { P(Amount won)  }  &  \frac { 1 } { 4 }  &  \frac { 1 } { 4 }  &  \frac { 1 } { 2 }\end{array}  Amount won   P(Amount won)   ​ $80 4 1 ​ ​ $30 4 1 ​ ​ $0 2 1 ​ ​ ​

Lemar Johnson · Accepted Answer

Final Answer : C, Explanation :  The probability model for this game can be constructed using a tree diagram as follows:- The first level represents the first draw, with a probability of 1/4 for each suit (since there are four suits in a deck).- If a club is drawn on the first attempt (probability 1/4), the player wins $80.- If a club is NOT drawn on the first attempt (probability 3/4), the game moves to the second level, representing the second draw. At this level, the probability of drawing a club is again 1/4, since the deck was replaced.- If a club is drawn on the second attempt (probability 1/4 x 1/4 = 1/16), the player wins an additional $30, for a total of $110.- If a club is NOT drawn on the second attempt (probability 1 - 1/16 = 15/16), the player wins nothing.Using this probability model, we can calculate the expected value of the game as follows:EV = (1/4)($80) + (3/4)(1/4)($30) + (3/4)(15/16)(0)EV = $20 + $2.81EV = $22.81Therefore, the best choice would be C, which represents the expected value of $22.81.

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