The Language of Argument

2 6 5

E x p e c t e d M o n e t a r y V a l u e

Compute the probability and the expected monetary value for the following

bets. Each time, you lay down $1 to bet that a certain kind of card will appear

from a standard fifty-two-card deck. If you win, you collect the amount

indicated, so your net gain is $1 less. If you lose, of course, you lose your $1.

Example: Draw a seven of spades. Win: $26.

Probability of winning 5 1/52

Expected value: [1/52 3 $(26−1)] 2 (51/52 3 $1) 5 2 $0.50

1. Draw a seven of spades or a seven of clubs. Win: $26.


2. Draw a seven of any suit. Win: $26.


3. Draw a face card (jack, queen, or king). Win: $4.


4. Do not draw a face card (jack, queen, or king). Win: $2.


5. On two consecutive draws without returning the first card to the deck,
   draw a seven of spades and then a seven of clubs. Win: $1,989.


6. Same as in problem 5, but this time the card is returned to the deck and
   the deck is shuffled before the second draw. Win: $1,989.


7. On two consecutive draws without returning the first card to the deck, do
   not draw a club. Win: $1.78.


8. Same as in problem 7, but this time the card is returned to the deck and
   the deck is shuffled before the second draw. Win: $1.78.


9. On four consecutive draws without returning any cards to the deck, a
   seven of spades, then a seven of clubs, then a seven of hearts, and then
   seven of diamonds. Win: $1,000,001.


10. On four consecutive draws without returning any cards to the deck, draw
    four sevens in any order. Win: $1,000,001.

Exercise I

Fogelin’s Palace in Border, Nevada, offers the following unusual bet. If you

win, you make a 50 percent profit on your bet; if you lose, you take a 40 percent

loss. That is, if you bet $1 and win, then you get back $1.50; if you bet $1 and

lose, you get back $0.60. The chances of winning are fifty-fifty. This sounds

like a marvelous opportunity, but there is one hitch: To play, you must let your

bet ride with its winnings, or losses, for four plays. For example, with $100,

a four-bet sequence might look like this:

__________________________________________________________Win Win Lose Win

Total $150 $225 $135 $202.50

At the end of this sequence, you can pick up $202.50, and thus make a $102.50

profit. It seems that Fogelin’s Palace is a good place to gamble, but consider

Exercise II

(continued)

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