1-4. 독립사상 문제풀이

1.4-1. Let A and B be independent events with P(A) =0.7 and P(B) = 0.2.

Compute

(a) P(A∩ B),

(b) P(A∪ B),

and (c) P(A'∪ B').

 

(a) = 0.7 x 0.2 = 0.14

(b) 0.7+0.2-0.14 = 0.76

(c) 1-0.14=0.86

 

1.4-2. Let P(A) = 0.3 and P(B) = 0.6. 

(a) Find P(A ∪ B) when A and B are independent.

(b) Find P(A|B) when A and B are mutually exclusive.

 

(a) 0.3+0.6 -0.18 = 0.72

(b) P(A|B) = P(A∩ B)/P(B) =0/0.6 (P(A∩ B)=0)

 

 

1.4-3. Let A and B be independent events with P(A) =1/4 and P(B) = 2/3.

Compute

(a) P(A∩B), 1/4 x 2/3 =1/6

(b) P(A∩B'), 1/4 x 1/3 = 1/12

(c) P(A'∩ B'), = 3/4 x 1/3 = 1/4

(d) P[(A ∪ B') ] 1/4 + 1/3 - 1/12 = 1/2

and (e) P(A'  ∩ B). = 3/4 x 2/3 = 1/2

 

 

 

1.4-4. Prove parts (b) and (c) of Theorem 1.4-1. 

증명은 생략

 

1.4-5. If P(A) = 0.8, P(B) = 0.5, and P(A ∪ B) = 0.9, are A and B independent events?Why or why not?

if independent P(A∩B)=P(A)P(B),

P(AUB) = P(A) +P(B) - P(A∩B)

0.9 = 0.8 + 0.5 -0.4

이는 독립이다.

 

1.4-6. Show that if A, B, and C are mutually independent, then the following pairs of events are independent: A and

(B ∩ C), A and (B ∪ C), A  and (B ∩ C ). Show also that A , B , and C  are mutually independent.

 

1.4-7. Each of three football players will attempt to kick a field goal from the 25-yard line. Let Ai denote the event that the field goal is made by player i, i = 1, 2, 3. 

Assume that A1, A2, A3 are mutually independent and that P(A1) = 0.5, P(A2) = 0.7, P(A3) = 0.6.

(a) Compute the probability that exactly one player is successful.

P(A1 ∩ A2' ∩ A3') = 0.5 x 0.3 x 0.4 =0.06

P(A1' ∩  A2 ∩  A3') = 0.5 x 0.7 x 0.4 = 0.14

P(A1'  ∩  A2' ∩  A3) = 0.5 x 0.3 x 0.6 =  0.09

0.29

(b) Compute the probability that exactly two players make a field goal (i.e., one misses).

P(A1 ∩ A2  ∩  A3') =0.5 x 0.7 x 0.4

P(A1 ∩ A2' ∩ A3) = 0.5 x 0.3 x 0.6

P(A1' ∩ A2  ∩ A3) = 0.5 x 0.7  x 0.6)

 

 

1.4-8. Die A has orange on one face and blue on five faces, Die B has orange on two faces and blue on four faces, Die C has orange on three faces and blue on three faces. All are fair dice. If the three dice are rolled, find the probability that exactly two of the three dice come up orange.

 

P(A)=1/6

P(B) = 2/6

P(C) = 3/6

 

P(A ∩ B C') = 1/6 x 2x6 x 3/6

P(A  ∩ B' ∩ C) = 1/6 x 4/6 x 3/6

P(A' ∩ B ∩ C) = 5/6 x 2/6 x 3/6

 

1.4-9. Suppose that A, B, and C are mutually independent events and that P(A) = 0.5, P(B) = 0.8, and P(C) = 0.9. Find the probabilities that (a) all three events occur, (b) exactly two of the three events occur, and (c) none of the events occurs.

1.4-10. Let D1, D2, D3 be three four-sided dice whose sides have been labeled as follows:

D1 : 0333 D2 : 2225 D3 : 1146

The three dice are rolled at random. Let A, B, and C be the events that the outcome on die D1 is larger than the outcome on D2, the outcome on D2 is larger than the outcome on D3, and the outcome on D3 is larger than the outcome on D1, respectively. Show that (a) P(A) = 9/16, (b) P(B) = 9/16, and (c) P(C) = 10/16.

Do you find it interesting that each of the probabilities  that D1 “beats” D2, D2 “beats” D3, and D3 “beats” D1 is greater than 1/2? Thus, it is difficult to determine the “best” die.

1.4-11. Let A and B be two events.

(a) If the events A and B are mutually exclusive, are A and B always independent? If the answer is no, can

they ever be independent? Explain. (b) If A ⊂ B, can A and B ever be independent events?

Explain.

1.4-12. Flip an unbiased coin five independent times. Compute the probability of

(a) HHTHT.

(b) THHHT.

(c) HTHTH.

(d) Three heads occurring in the five trials.

1.4-13. An urn contains two red balls and four white balls. Sample successively five times at random and with replacement, so that the trials are independent. Compute the probability of each of the two sequences WWRWR and RWWWR.

1.4-14. In Example 1.4-5, suppose that the probability of failure of a component is p = 0.4. Find the probability

that the system does not fail if the number of redundant components is

(a) 3.

(b) 8.

1.4-15. An urn contains 10 red and 10 white balls. The balls are drawn from the urn at random, one at a time.

Find the probabilities that the fourth white ball is the fourth, fifth, sixth, or seventh ball drawn if the sampling

is done

(a) With replacement.

(b) Without replacement.

(c) In the World Series, the American League (red) and National League (white) teams play until one team

wins four games.Do you think that the urn model presented in this exercise could be used to describe the

probabilities of a 4-, 5-, 6-, or 7-game series? (Note that either “red” or “white” could win.) If your answer

is yes, would you choose sampling with or without replacement in your model? (For your information,

the numbers of 4-, 5-, 6-, and 7-game series, up toand including 2012, were 21, 24, 23, 36. This ignores

games that ended in a tie, which occurred in 1907,1912, and 1922. Also, it does not include the 1903 and

1919–1921 series, in which the winner had to take five out of nine games. The World Series was canceled in

1994.)

1.4-16. An urn contains five balls, one marked WIN and four marked LOSE. You and another player take turns

selecting a ball at random from the urn, one at a time. The first person to select the WIN ball is the winner. If

you draw first, find the probability that you will win if the sampling is done

(a) With replacement.

(b) Without replacement.

1.4-17. Each of the 12 students in a class is given a fair 12-sided die. In addition, each student is numbered from 1 to 12.

(a) If the students roll their dice, what is the probability that there is at least one “match” (e.g., student 4 rolls a 4)?

(b) If you are a member of this class, what is the probability that at least one of the other 11 students rolls the

same number as you do?

1.4-18. An eight-team single-elimination tournament is set up as follows: