With 3 plant cuttings inserted in each of 100 pots and with 60% rooting, how many pots will have all 3 rooted, only 2 rooted, only 1 rooted, and none rooted?

Question

In plant propagation we insert parts of shoots of plants (called cuttings) into soil to root them. Our job as plant propagators is to grow roots on these cuttings. Once a cutting is rooted, it can be grown to a larger plant size and sold from a nursery.

In this nursery production example, we insert 3 cuttings in each nursery pot. From experience, 60 out of 100 cuttings inserted will root so the rooting percentage = 60%

If 100 pots each have 3 cuttings inserted initially, how many pots will have all 3 cuttings rooted, only 2 cuttings rooted, only 1 cutting rooted, and no cuttings rooted using the 60% rooting percentage.

Answer 1

The probability that any cutting takes root is 0.60. Either it takes root, or it doesn't--a binomial distribution. So for each cutting, p = 0.6 and q = p^-1 = 1 - p = 0.4. So for each pot, let n = 3. Iterating over the cuttings in each pot using the BInomial Distribution Formula:

P(n, k, p) = C(n,k) p^k * (1 - p)^(n-k),

where n is umber of trials, k is number of desired outcomes and p is the probability of that outcome in independent trials. So...

The probability of k = 0 successes in n = 3 trials while probability of success is p = 0.6 is

C(3,0) * ( (0.6) ^ 0) * ((0.4)^(3 - 0)) = 0.064

The probability of k = 1 successes in n = 3 trials while probability of success is p = 0.6 is

C(3,1) * ( (0.6)^1) * ((0.4)^(3 - 1)) = 0.288

The probability of k = 2 successes in n = 3 trials while probability of success is p = 0.6 is

C(3,2) * ( (0.6)^2) * ((0.4)^(3 - 2)) = 0.432

The probability of k = 3 successes in n = 3 trials while probability of success is p = 0.6 is

C(3,3) * ( (0.6)^3) * ((0.4)^(3 - 3)) = 0.216

(Notice also that the you can check your work by verifying that the TOTAL PROBABILITY for this distribution over [0:n] is

(k=0,k=n) ∑ C(n,k) p^k * (p-1)^(n-k)

= 0.064 + 0.288 + 0.432 + 0.216

= 1.)

So for 100 pots, we would expect 100*0.064 = 6.4 --> 6 pots to have 0 rooted cuttings,

100 * 0.288 = 28.8 -- > 29 pots to have exactly 1 rooted cutting,

100 * 0.432 = 43.2 --> 43 pots to have exactly 2 rooted cuttings and

100 * 0.216 = 21.6 --> 22 pots to have exactly 3 rooted cuttings.