(a) An association is made up of 6 farmers and 8 traders. If an executive body of 4 members is to be formed, find the probability that it will consist of at...

(a) An association is made up of 6 farmers and 8 traders. If an executive body of 4 members is to be formed, find the probability that it will consist of at least two farmers. (b) The probability of an accident occurring in a given month in factories X, Y, and Z are \(\frac{1}{5}, \frac{1}{12} \) and \(\frac{1}{6}\) respectively.

Find the probability that the accident will occur in:

i) none of the factories;

(ii) all the factories;

(iii) at least one factory.

Answer Details

(a) To find the probability that the executive body will consist of at least two farmers, we need to calculate the probability of two farmers, three farmers, or four farmers being selected.

The total number of possible executive bodies of 4 members is given by:
n(S) = (6+8) choose 4 = 3003

The number of ways to select 2 farmers and 2 traders is:
n(A) = (6 choose 2) x (8 choose 2) = 420

The number of ways to select 3 farmers and 1 trader is:
n(B) = (6 choose 3) x (8 choose 1) = 160

The number of ways to select 4 farmers is:
n(C) = (6 choose 4) = 15

Therefore, the probability of selecting an executive body of 4 members that consists of at least two farmers is:
P(A or B or C) = (420 + 160 + 15) / 3003 = 0.297

So, the probability is 0.297 that the executive body will consist of at least two farmers.

(b) (i) The probability that an accident will occur in none of the factories is:
P(none) = (4/5) x (11/12) x (5/6) = 0.4583

(ii) The probability that an accident will occur in all three factories is:
P(all) = (1/5) x (1/12) x (1/6) = 0.0007

(iii) To find the probability that an accident will occur in at least one factory, we need to find the probability of the complement event (i.e., no accidents in any of the factories) and subtract it from 1:
P(at least one) = 1 - P(none) = 1 - 0.4583 = 0.5417

So, the probability is 0.4583 that no accidents will occur in any of the factories, the probability is 0.0007 that accidents will occur in all the factories, and the probability is 0.5417 that at least one accident will occur in any of the factories.