**Answer:**

a) **The probability mass function of K = **

**b) **

**c) **

**Step-by-step explanation:**

a) Let p be the probability of winning each ticket be = 0.1

Then q which is the probability of failing each ticket = 1 - p = 1 - 0.1 = 0.9

Assume X represents the number of failure preceding the 5th success in x + 5 trials.

The last trial must be success whose probability is p = 0.1 and in the remaining (x + r- 1) ( x+ 4 ) trials we must have have (4) successes whose probability is given by:

Then, the probability distribution of random variable X is

where;

X represents the negative binomial random variable.

K= X + 5 = number of ticket buy up to and including fifth winning ticket.

Since K =X+5 this signifies that X = K-5

as X takes value 0, 1 ,2,...

K takes value 5, 6 ,...

Therefore:

**The probability mass function of K = **

b)

Let p represent the probability of getting a tail on a flip of the coin

Thus p = 0.5 since it is a fair coin

where L = number of flips of the coin including 33rd occurrence of tails

Thus; the negative binomial distribution of L can be illustrated as:

where

X= L

r = 33 &

p = 0.5

Since we are looking at the 33rd success; L is likely to be : L = 33,34,35...

Thus; the PMF of L =

c)

Given that:

Let M be the random variable which represents the number of tickets need to be bought to get the first success,

also success probability is 0.01.

Therefore, M ~ Geo(0.01).

Thus, the PMF of M is given by: