Question

(a) Suppose a dealer draws one card from a standard, properly shuffled, 52-card deck of cards (that is, all Jokers have been removed). (i) Describe the sample space - i.e. all of the possible outcomes (no need to write them all out, just describe them in words). (1 point) (ii) What is the probability that the card is a Heart? (1 point) (iii) What is the probability that the card is a 6? (1 point) (iv) What is the probability that the card has a number on the face (i.e. not Jack, Queen, King, or Ace)? (1 point)

(b) Suppose I flip a fair coin twice in a row. (i) Write out the set of possible outcomes (the sample space). (1 point) (ii) What is the probability that at least one flip lands with Heads facing up? (1 point) (iii) What is the probability that both flips land with Heads facing up? (1 point)

(c) Consider a business owner who can make one of three decisions about a new product. Each decision generates a lottery over different possible revenue outcomes. If he makes Decision A, then the resulting lottery, p A, generates $20, 000 in revenue with probability 0.3, $12, 000 in revenue with probability 0.15, $5, 000 in revenue with probability 0.35, and $2, 000 with probability 0.2. If he makes Decision B, then the resulting lottery, p B, generates $30, 000 in revenue with probability 0.1, $10, 000 in revenue with probability 0.6, and $1, 000 with probability 0.3. If he makes Decision C, then the resulting lottery, p C , generates $20, 000 in revenue with probability 0.2, and $8, 000 with probability 0.8. (i) Write each lottery in the form (p1, x1; p2, x2; ...; pn, xn). (2 points) (ii) Calculate the expected value (of revenue) from each decision. (3 points)

Answer 1

A. A deck of 52 cards is shuffled..

1. The sample space would be 52. It includes all the hearts, spades, diamonds( each 13 cards) & clubs of Red and Black color.

2. Total number of hearts in a deck = 13

Total Number of cards = 52

Probability of hearts = 13/52 = 1/4

3. There are four 6s in a deck of 52 cards. One each of Heart, Diamond, Club & Spade.

Therefore, probability of 6s in a deck = 4/52 =1/13

4. There are 4 cards each of Ace, Jack, king and Queen in the deck. That makes it a total of 16 cards in the deck of 52 cards. Rest all of the cards are numbered from 2 to 9.

Total number of cards = 52

Ace, Jack, king and Queen all put together = 16

Therefore, total number of Numbered cards = 52-16 = 36

Probability of Numbered cards = 36/52 = 9/13

B. A coin is flipped twice in a row.

1. The total number of outcomes would be 2ⁿ = 2² = 4

2.  Probability that atleast one flip lands with heads = 1 - Probability that none of the flips lands with heads

Probability that none of the flips lands with heads =  Probability that both the flips lands with tails

Therefore, Probability that atleast one flip lands with heads = 1 - Probability that both the flips lands with tails

Probability that both the flips lands with tails :

P( Tail) = P( Heads) = 0.5

Probability that both the flips lands with tails = * P(Tail)ⁿ * P(Head)^2-n ( Where n = no of tails)

=   * P(Tail)² * P(Head)^2-2   =    * (0.5)² * (0.5)⁰ = 0.25

Therefore, Probability that atleast one flip lands with heads = 1 - Probability that both the flips lands with tails

= 1 - 0.25 = 0.75

3. Probability that both the flips lands with heads :

Probability that both the flips lands with heads= * P(Tail)^2-n * P(Head)ⁿ ( Where n = no of heads)

=   * P(Tail)⁰ * P(Head)² =    * (0.5)² * (0.5)⁰ = 0.25

C.

1.

Decision A
Revenue	Prob
$      20,000	0.3
$      12,000	0.15
$        5,000	0.35
$        2,000	0.2

Decision B
Revenue	Prob
$ 30,000	0.1
$ 10,000	0.6
$   1,000	0.3

Decision C
Revenue	Prob
$ 20,000	0.2
$   8,000	0.8

Expected Revenue at A = 20000*0.3 + 12000*0.15 + 5000*0.35 + 2000*0.2 = $9950

Expected Revenue at B = 30000*0.1 + 10000*0.6 + 1000*0.3 = $9300

Expected Revenue at C = 20000*0.2 + 8000*0.8 = $10400