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

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

Quizplus · Accepted Answer

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 + $2.81
EV = $22.81

Therefore, the best choice would be C, which represents the expected value of $22.81.

You Pick a Card from a Deck B) C) D)