Complete and submit solutions to the following problems. Define events properly

Question

Complete and submit solutions to the following problems. Define events properly and use correct notation for the probabilities you use to determine your answers. Show all your work in detail. Answers without supporting work may be severely penalized 1. Problem 3 in your textbook presents three different sample spaces for a horse race with five horses running. Let the horses be designated by the letters A, B, C, D, and E. A more complete characterization of an outcome of the horse race would be to designate which horse finishes in first through fifth places. (a) Consider the sample space for the set of outcomes characterized in this way. How many such outcomes are in the sample space? b) How many outcomes are in the event that horse A finishes first? (c) If G is the event that horse A finishes first and H is the event that horse B does not finish second, describe in words the event GnH. How many outcomes are in this event? 2. Continuing with the five-horse race scenario, you determine that the probability of horse A winning the race is 0.4 based on previous performances in horse races. Horses B and A have raced in several races before, and you notice that in 30% of their common races, if horse A wins, horse B comes in second. Based on this information, what is the probability that horses A and B finish first and second, respectively? 3. Now suppose that each order of finish in the horse race is equally likely. (a) What is the probability that horse C wins the race? (b) What is the probability that horse D finishes last? (c) What is the probability that horse C wins and horse D finishes last? (d) Are the events "horse C wins" and "horse D loses" independent? Explain why or why not

Explanation / Answer

1) as there are 5 horses

A,B,C, D,E

a)they are running in a race so would be 5! (five factorial) possibilities of the position of horses when the race is finished.

thus there are 5! =5*4*3*2*1 = 120 outcomes

b) when A finishes first ,the position of A is fixed but other 4 can change their place

thus 4! =4*3*2*1 = 24 outcomes are possible

c) there will be three events that a finishes first and does not finish 2nd i.e. he can take 3rd, 4th 5th position

so 24-3=21 outcomes

GH=21 (G intersection H)

2) probability of A winning the race=0.4 =40%

probability that A will come first and B second =0.4*0.3=0.12

probability that horses a and B will complete the race first and second respectively =0.4+0.12=0.52

3) a) C wins the race =4!/ total number of outcomes (because the position of C is fixed)

=24/120= 0.2

b) D finishes last (position of D is fixed)

number of possibilities = 4!=24

probability=0.2

c) now the position of horse C and horse D is fixed so total no. of possibilities= 3! =6

probability= 6/120 =0.05

d) No, both the events are not independent of each other. if horse C wins horse D has to lose .it has no freedom . both horses can not win or lose in a race.

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.