Here, A1, A2, and A3 denote the three types of flower seeds.
and A1: A2: A3 = 4: 4 : 2
Total outcomes = 10
Let E be the event that a seed germinates and E’ be the event that
a seed does not germinate.
Now P(E|A1) is the probability that seed germinates when it is seed A1.
P(E’|A1) is the probability that seed will not germinate when it is seed A1.
P(E|A2) is the probability that seed germinates when it is seed A2.
P(E’|A2) is the probability that seed will not germinate when it is seed A2.
P(E|A3) is the probability that seed germinates when it is seed A3.
P(E’|A3) is the probability that seed will not germinate when it is seed A3.
The Law of Total Probability:
In a sample space S, let E1, E2, E3……. En be n mutually exclusive and exhaustive events associated with a random experiment. If A is any event which occurs with E1, E2, E3……. En, then
P(A) = P(E1) P(A|E1) + P(E2) P(A|E2) + …… P(En)P(A|En)
(i) Probability of a randomly chosen seed to germinate.
It can be either seed A, B or C.
So,
From law of total probability,
P(E) = P(A1) × P(E|A1) + P(A2) × P(E|A2) + P(A3) × P(E|A3)
(ii) that it will not germinate given that the seed is of type A3
As we know P(A) + P(A’) =1
P(E’|A3) = 1 – P(E|A3)
= 1 - 35/100
= 65/100
(iii) that it is of the type A2 given that a randomly chosen seed does not germinate.
We use Bayes’ theorem to find the probability of occurrence of an event A when event B has already occurred.
