Practical Class 01 PDF

Preview

Summary

CLASS PRACTICE
...

Description

SMS

STAT2062: Fundamentals of Probability with Applications

UWA

Question 1. Let A, B , C be three events. Fill in the following dictionary Expression A∩B∩C ? ? (A ∩ B c ) ∪ (Ac ∩ B) ? ?

Interpretation all three events A, B and C occur none of A, B and C occur A and B occur but not C ? at least one of A, B or C occurs exactly one of A, B or C occurs

Question 2. In each of the following experiments, identify the sample space S and also the event A mentioned: (a) a three-digit number is formed by arranging the digits 1, 5 and 6 in a random order. A is the event that the number is larger than 400. (b) a trainee pilot lands an aircraft on a runway 3 km long and the point of touchdown is noted. A is the event that the pilot touches down correctly, which is defined in the manual as touching down no more than 1 km from the start of the runway. (When answering this question you may assume that the trainee pilot will always land somewhere on the runway.) Question 3. Suppose a fair die has its even-numbered faces painted red and the odd-numbered faces are white. Consider the experiment of rolling the die once and the events A = {2 or 3 shows up} and B = {a red face shows up}. Find the following probabilities. (a) P (A) (b) P (B ) (c) P (A ∩ B ) (d) P (A|B ) (e) P (A ∪ B ) Question 4. A card is drawn at random from a standard pack of 52 cards. Let A be the event that the card is a King and B the event that it is a Heart. (a) are A and B mutually exclusive? (b) are A and B independent? Explain your answers. Reminder: A standard pack of 52 cards consists of 4 suits with 13 cards each. The 4 suits are Heart (♥), Diamond (♦), Clubs (♣) and Spade (♠). Each suit consists of 13 cards with face values: 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King and Ace. Question 5. Individual A has a circle of five close friends (B, C, D, E and F). A has heard a certain rumour from outside the circle and has invited the five friends to a party in order to circulate the rumour. To begin, A selects one of the five at random and tells the rumor to the chosen individual. That individual then selects at random one of the four remaining individuals and repeats the rumour. Continuing, a new individual is selected from those not already having heard the rumour by the individual who has just heard it, until everyone has been told. (a) What is the probability that the rumor was repeated in the order B, C, D, E and F? Semester 2, 2017

1

Questions for week 2

SMS

STAT2062: Fundamentals of Probability with Applications

UWA

(b) What is the probability that F is the third person at the party to be told the rumour? (c) What is the probability that F is the last person to hear the rumour? Clearly state any assumptions that you make when answering this question. Question 6.

(a) Let A and B be two events.

(1) Show that P (A ∩ B) ≤ P (A) and P (A ∪ B) ≥ P (A). (2) Show that if B ⊆ A then P (B) ≤ P (A). (b) Show that if events A and B are independent then A and B c are independent too. (c) Show that events A and B are independent if P (A|B) = P (A|B c ). Question 7. Assume that we roll a fair die twice and that the outcomes are independent of each other. Let A be the event that the first roll showed three or less, B that the second roll showed three or less and C that both rolls showed either both less than four or both more than three. (a) Show that A and B are independent, A and C are independent, and B and C are independent. (b) Show that A, B and C are not independent. Question 8. A certain company send 40% of its overnight mail parcels via express mail service E1 , 50% via express mail service E2 and the remaining 10% are sent via E3 . Of these parcels, those sent via E1 2% arrive after the guaranteed delivery time (denote by L the event “late delivery”). Of those sent via E2 , only 1% arrive late, whereas 5% of the parcels handled by E3 arrive late. (a) If a record of an overnight mailing is randomly selected from the company’s files, what is the probability that the parcel went via E1 and was late? (b) What is the probability that a randomly selected parcel arrived late? (c) If a randomly selected parcel has arrived on time, what is the probability that it was not sent via E1 ? Question 9. A particular airline has 10:00am flights from Perth to Adelaide, Melbourne and Sydney. Let A denote the event that the Adelaide flight is full and define events B and C analogously for the other two flights. Suppose P (A) = 0.6, P (B ) = 0.5, P (C ) = 0.4 and the three events are independent. What is the probability that (a) All three flights are full? That at least one flight is not full? (b) Only the Adelaide flight is full? That exactly one of the three flights is full? Question 10. The Belgium 20-franc coin (B20), the Italian 500-lire coin (I500) and the Hong Kong 5-dollar coin (H K5) are approximately of the same size. Coin purse 1 (C 1) contains 6 of each of these coins. Coin purse 2 (C2) contains 9 B20s, 6 I500s and 3 H Ks. A fair four-sided die is rolled. If the outcome is {1}, a coin is selected randomly from C1. If the outcome belongs to {2, 3, 4}, a coin is selected randomly from C2. Find Semester 2, 2017

2

Questions for week 2

SMS

STAT2062: Fundamentals of Probability with Applications

UWA

(a) P (B20), the probability of selecting a Belgian coin. (b) P (C1|B20), the probability that the coin was selected from C1, given that it was a Belgian coin. Question 11. In the seventeenth century the Chevalier de M´er´e, a French nobleman and gambler, started a controversy by claiming that it should be just as likely to get at least one six in four throws of a die as it is to get at least one double six in 24 throws of two dice. Is de M´er´e’s claim correct? If so, why? If not, which is more likely? Question 12. Assume we deal a hand of 13 cards from a well-shuffled standard pack of 52 cards (i.e. every possible hand is equally likely to be dealt) (a) What is the probability that in a player’s hand of 13 cards at least one suit will be missing? (b) What is the probability that a hand of 13 cards contains (at least one) four of a kind? Question 13. Let X be the number of tires on a randomly selected automobile which are underinflated. (a) Which of the following three pX (x) functions is a legitimate p.m.f. for X, and why are the other two not allowed? x pX (x) pX (x) pX (x)

0 0.3 0.4 0.4

1 0.2 0.1 0.1

2 0.1 0.1 0.2

3 0.05 0.1 0.1

4 0.05 0.3 0.3

(b) For the legitimate p.m.f. of part (a), compute P (2 ≤ X ≤ 4), P (X ≤ 2) and P (X 6= 0). (c) If the p.m.f. for X has the form pX (x) = c · (5 − x) for x = 0, 1, . . . , 4, what is the value of c? Question 14. For each of the following, determine the constant c so that p(x) satisfies the conditions of being a p.m.f. for a random variable X . (a) p(x) = c(1/5)x , x = 1, 2, 3, . . . (b) p(x) = cx2 , x = −3, −2, . . . , 3. Question 15. Suppose that a box contains seven red balls and three blue balls. If five balls are selected at random, without replacement, determine the probability mass function of the number of red balls that will be obtained.

Semester 2, 2017

3

Questions for week 2...