Question

a swindler at the local carnival has a gaming booth designed to dupe unsupervised children with poor math skills, according to cards drawn from a deck. cassidy has observed several of her friends play the game, and using her knowledge of card decks and probability theory, she has estimated the following information:

card draw	probability	winnings or losses
card #2	0.21	$0.60
card #3	0.82	$0.25
card #4	0.54	-$0.25

it costs $1.00 to play the game and draw all four cards. the first card has a 7% chance of the player winning $6.00, but if that does not happen then there is no financial gain or loss. the second card has a 21% chance of the player winning $0.60, and so on according to the table. what are the expected winnings for someone who draws all four cards, not counting the $1.00 they paid to play, and rounded to the nearest penny?

blank dollars

question help: ▶ video

Explanation:

Step1: Recall Expected Value Formula

The expected value \( E(X) \) of a discrete random variable is calculated as \( E(X)=\sum_{i = 1}^{n}x_{i}P(x_{i}) \), where \( x_{i} \) is the value of the random variable and \( P(x_{i}) \) is the probability of that value.

Step2: Calculate Each Term

• For card #1: \( x_1 = 6.00 \), \( P(x_1)=0.07 \), so the term is \( 6.00\times0.07 = 0.42 \)
• For card #2: \( x_2 = 0.60 \), \( P(x_2)=0.21 \), so the term is \( 0.60\times0.21=0.126 \)
• For card #3: \( x_3 = 0.25 \), \( P(x_3)=0.82 \), so the term is \( 0.25\times0.82 = 0.205 \)
• For card #4: \( x_4=- 0.25 \), \( P(x_4)=0.54 \), so the term is \( - 0.25\times0.54=-0.135 \)

Step3: Sum the Terms

Add up all the terms: \( 0.42 + 0.126+0.205- 0.135 \)
First, \( 0.42+0.126 = 0.546 \)
Then, \( 0.546 + 0.205=0.751 \)
Then, \( 0.751-0.135 = 0.616 \)

Step4: Round to Nearest Penny

Rounding \( 0.616 \) to the nearest penny (two decimal places) gives \( 0.62 \)

Answer:

\( 0.62 \)