Statistics and probability distribution

Question

Hailstorms are common phenomena in parts of a certain country. From past experience it has been shown that in a forestry plantation, there is about an 80% chance that certain tree species will survive the storm. Suppose three (3) trees are selected at random. Find

i. Probability that the number of trees surviving the hailstorm will be exactly two                                                                                                 [2 marks]

ii. Probability that the number of tree surviving the hailstorm will be at most one.                                                                                                 [5 marks]                                                       

iii. Probability that the number of tree surviving the hailstorm will be at least one.                                                                                                 [5 marks]                                                                                      

iv. Expected number of trees that will survive the hailstorm. [1 mark

Answer 1

P(x = k), where k = number of surviving trees

P(x = k) = n! p^k * (1-p)^(n-k)

--------

k! (n-k)!

n = 3

p = 0.8

i)

P(x=2 ) = 3! (0.8)^2 * (0.2)^1

--------

2! 1!

ii) P(x = 0 or = 1) = P(x = 0) + P(x = 1) - use formula as per i)

iii) P(at least 1) = 1 - P(x = 0) - use formula as per i)

iv) expected value = np = 3 * 0.8 = 2.4