7. (13 total points) Consider a state lottery where five numbers between 1 and 4

Question

7. (13 total points) Consider a state lottery where five numbers between 1 and 43 are drawn. Note that the order in which numbers appear on a ticket does not matter (for instance, a ticket with 12, 14,41, 7,29 would be the same as a ticket with 29, 12,41, 7, 14). There are two possible winning scenarios: match all five numbers and match any three numbers. You do not need to solve the actual values for any of the parts for this question You cam leave answers written as factorials and/or as fractions. For example. if you are doing a permutation of 10 permute 6, you can write (30-5 (10-6) (a) (2 points) How many possible tickets are there? (b) (2 points) What is the probability of matching all five numbers? (c) (4 points) What is the probability of matching any three numbers? (d) (5 points) Given that a ticket matches three numbers, what is the probability of matching all five numbers? 8. (2 points Extra Credit) Show that the exponential distribution has the memoryless property. In other words, show Px > a + blx> a) P(x> b) for constants a and b

Explanation / Answer

a)

Total Possible Tickets = 43C5 = 962598

b)

Probability = Favourable Outcomes/Total Outcomes = 120/962598 = 0.00012

Favourable Outcomes eg:

12 14 41 7 29

=5*4*3*2*1

c)

Favourable Outcomes = 38C2 * 5C3 = 7030

Probability = 7030/962598 = 0.0073

d)

Total Outcomes = 38C2 * 5C3 = 7030

Favourable Outcomes = 5C5 * factorial(5)   120

Probability = 120/7030 = 0.01707

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