Chapter 12: Q. AP4.32 (page 832)

AP4.32 A standard deck of playing cards contains $52$ cards, of which $4$ are aces and $13$ are hearts. You are offered a choice of the following two wagers:
I. Draw one card at random from the deck. You win $$10$ if the card drawn is an ace. Otherwise, you lose $$ 1$ .
II. Draw one card at random from the deck. If the card drawn is a heart, you win $$2$ . Otherwise, you lose $$ 1$ . Which of the two wagers should you prefer?
a. Wager $1$ , because it has a greater expected value
b. Wager $2$ , because it has a greater expected value
c. Wager $1$ , because it has a greater probability of winning
d. Wager $2$ , because it has a greater probability of winning
e. Both wagers are equally favorable

Short Answer

Expert verified

The correct answer is option (a) Wager $1$ , because it has a greater expected value

Step by step solution

Given information

To determine a choice of the two wagers by draw one card at random from the deck.

Explanation

Let, the given option of selecting one of the steps listed below.
If we draw an ace, will win $$ 10$ ; otherwise, will lose $$ 1$ , and if we draw a heart, will win $$ 2$ ; otherwise, will lose $$ 1$ .
The amount of each possibility multiplied by its probability is the estimated value:
Calculate for wager $1$ as follows:
$E (X) = \sum x P (x)$
$= $ 10 \times \frac{4}{52} + (- $ 1) \times \frac{48}{52}$
$= - $ 0.15$