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 can learve answers writen as factorials and or as fractions. For example. if you are doing a permutation of 10 permute 6, you can write (0-0) (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? (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. 8.

Explanation / Answer

Ans for 7/a)

There are 43 numbers and only 5 of them will be on ticket hence we will have to use Combination concept here answer will be 43C5=43!/(38!*5!) ticket combinations are possible

Answer for 7/b)

There will be 5*4*3*2*1 winning tickets if order is not the concern then probability of matching all the numbers on ticker will be

Possible number of winning tickets/Total number of tickets=120*(38!*5!)/43!

Ans for 7/c)

Probability of Matching any 3 numbers will be 5*4*3*40*39*38!*5!/43!

5*4*3*40*39 means one number out of 5 numbers that are chosen on lottary then 4 and finally 3 numbers to choose from where we can choose only one number after that 40 and 39 are remaining numbers from a slot this multiple is divded by number of tickets possible

ANs 7/d)

If one ticker is matched with 3 numbers and to match 5 numbers then probability will be 5*4*3*2*1*38!*5!/43!

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