The probability that a clover has four leaves is .2. Kimber and Janice have gone clover hunting in a field...?

Question

full of clovers.
a) Let T be the total number of clovers picked until they each pick their first 4-leaf clover. What is the distribution of T?

b) Let S be the number of clovers picked by Janice until her 3rd 4-leaf clover, what is the distribution of S?

c)Suppose Janice's 5th clover was her 3rd 4-leaf clover. Let U be the number of subsequent clovers picked until her next 4-leaf clover. What is the distribution of U?

d)If I tell you that since having found her 3rd 4-leaf clover, Kimber has picked another 10 and still not found another 4-leaf clover. Given this information what is the probability she picks at least another 15 until getting her next 4-leaf clover?

e)Let V be the total number of clovers picked by both of them until Janice finds her 1st and Kimber finds her 3rd 4-leaf clover. What is the distribution of V?

Answer 1

Dear ianboen86,

The clover picking described here is a Bernoulli process. Since the successes and failures are integers, a special case of the negative binomial distribution known as the Pascal distribution should be of the form that is suitable for your questions.

For notation, I will let p = 0.2 and q = 1 - p = 0.8; I'll leave it to you to substitute for p and q if you wish, but it seems unnecessary since the questions are not asking for you to compute numerical probabilities, except possibly in part d). Also C(n,k) is "n choose k," given as
C(n,k) = n! / [k! (n - k)!].

I will also take "until" as meaning up to AND including the events specified in the questions. For example, in the first question, I interpret T to be the total number of clovers picked including the first 4-leaf clover. (An alternative interpretation could be the number of clovers preceding, but not including, the 4-leaf clover of interest. If you choose that interpretation, then you will need to make slight adjustments to the distributions I show to reconcile them with this alternative interpretation.)

a) f(T) = p q^(T - 1), for T in {1, 2, 3, . . . }, and
f(T) = 0 otherwise.

b) f(S) = C(S - 1, 3 - 1) p^3 q^(S - 3)
= C(S - 1,2) p^3 q(S - 3), for S in {3, 4, 5, . . . }, and
f(S) = 0 otherwise.

c) Since each pick occurs independently, the history of the process tells us nothing about the future. Thus, the distribution of U for the number of clovers picked until the next 4-leaf clover is the same as it was for the first 4-leaf clover, given by the distribution of T, above (where the possible values of U are starting relative to the previous clover picked, rather than starting with the very first clover picked).

f(U) = p q^(U - 1), for U in {1, 2, 3, . . . }, and
f(U) = 0 otherwise.

d) Again, the history does not inform the future for this process, so we need only be concerned with whether Kimber picks a 4-leaf clover in the next 14 picks or not. Using the same distribution function as described for U in the previous question we find the following.

P(number of picks before next 4-leaf clover >= 15)
= 1 - P(number of picks before next 4-leaf clover < 14)
= 1 - [f(1) + f(2) + f(3) + . . . + f(14)]
= 1 - [p q^0 + p q^1 + p q^2 + . . . + p q^13]
= 1 - p [q^0 + q^1 + q^2 + . . . + q^13]
= 1 - 0.2 [4.78010]
= 1 - 0.95602
= 0.04398 (to five decimal places).

e) f(V) = C(V - 1,4 - 1) p^4 q^(V - 4)
= C(V - 1,3) p^4 q^(V - 4), for V in {4, 5, 6, . . . }, and
f(V) = 0 otherwise.