# Prob & expected value (40分!!)

1) What is the probability of not obtaining either an ace or a king in a single blind draw from a standard deck of cards.

2) What is the probability of obtaining either 4 tails or 4 heads in four tosses of a fair coin?

3) Suppose you throw two unbiased dice. What is the probability of the sum of the two numbers that the dice land on is less than 4?

4) Suppose there are 23 cards remaining from a standard deck of cards. 13 of the cards are diamonds, and 7 of the cards are hearts, and 4 of the cards are royals. If a card is blindly drawn from the pack of cards, what is the probability that it is a royal diamond?

5) Suppose you are walking to work when you notice that it looks like it is about to rain and that you have forgotten your umbrella. There are two actions you can do. You can either go home and get your umbrella, in which case you will need to get a taxi in order to get to work on time. Alternatively, you can keep walking to work and risk getting wet. The probability of it raining it 0.2, and if it rains, it will wreck your suit, which will cost you \$400.

a) What is the expected monetary gain of going home, getting your umbrella and getting the taxi to work?

b) What is the expected monetary gain of walking to work?

**第五條少左D野!! It should be part of the question that the taxi ride costs \$60

### 2 個解答

• 最愛解答

1) P(ace or king)=1/13+1/13=2/13

P(neither)=1-P(ace or king)=11/13

2) P(4 tails)=1/2*1/2*1/2*1/2=1/16

P(4 heads)=1/16 (procedure same as above)

P(either)=1/16*2=1/8

3)

2 3 4 5 6 7

3 4 5 6 7 8

4 5 6 7 8 9

5 6 7 8 9 10

6 7 8 9 10 11

7 8 9 10 11 12

P(<4)=3/36=1/12

4) Let me gather the information from the problem: 13 diamonds, 7 hearts, 3 spades or clubs. 4 of the above are royal.

Let R be royal and D be diamond.

We are looking for P(R^D).

P(R^D)=P(R)*P(D|R)=4/23*10/23=4/10

This problem can be easily solved if you draw a "tree". However, if you know Bayes well, it wouldn't be a hard one.

5) Case 1: go home to get umbrella, and then take taxi back to work

Case 2: go to work directly

Case 1 EV=1*(-60)=-60

Case 2 EV=.2*(-400)+.8*0=-80

Hope those answers help. However, it really took me a while to figure them out.

資料來源： myself~ I love prob&stats!
• 20點問咁多題好大機會無人答。

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