Grafting, the uniting of the stem of one plant with the stem or root of another, is widely used commercially to grow the stem of one variety that produces fine fruit on the root system of another variety. Suppose grafts are done one at a time. Each graft is successful with probability 0.55, independently of each other. Given there were no failed grafts in the first seven done, what is the probability that there will be at least one failed graft in the next four

Given that Grafting, the uniting of the stem of one plant with the stem or root of another, is widely used commercially to grow the stem of one variety that produces fine fruit on the root system of another variety. Suppose grafts are done one at a time. Each graft is successful with probability 0.55, independently of each other.

We have X no of failures also would be binomial with p = 0.45 since each is independent of other.

Required probability

= P(x>0 in next four trials/x=0 in I seven trials)

= P(x>0 in next four trials)*P(x=0 in I seven trials)/P(x=0 in I seven trials)

= 1-P(x=0) in 4 trials

= [tex]1-(0.45)^4\\=0.9590[/tex]