A chairman is biased so that he selects his relatives for a job 3 times as likely as others. If there are 3 posts for a job. Find the probability distribution for selection of persons other than
their relatives.
To find the probability distribution for the selection of persons other than the chairman's relatives, we need to determine the probability of each outcome.
Let's denote the probability of selecting a non-relative candidate as p and the probability of selecting a relative candidate as 3p (3 times more likely). Since there are 3 posts, we need to consider the possibilities of selecting 0, 1, 2, and 3 non-relative candidates.
1. Selecting 0 non-relative candidates:
The probability of selecting 0 non-relative candidates is (3p)^3, as we need to select all three relatives in this case.
2. Selecting 1 non-relative candidate:
The probability of selecting 1 non-relative candidate is 3(3p)^2 * p, as we have three ways to choose which position is occupied by a non-relative candidate.
3. Selecting 2 non-relative candidates:
The probability of selecting 2 non-relative candidates is 3(3p) * (3p)^2, as we have three ways to choose two positions for non-relative candidates.
4. Selecting 3 non-relative candidates:
The probability of selecting 3 non-relative candidates is (3p) * (3p)^2, as we have only one way to choose three positions for non-relative candidates.
To find the value of p, we need to set up an equation. Since the total probability must sum to 1, we have the following equation:
(3p)^3 + 3(3p)^2 * p + 3(3p) * (3p)^2 + (3p) * (3p)^2 = 1
Simplifying this equation:
27p^3 + 27p^3 + 27p^3 + 27p^3 = 1
108p^3 = 1
Dividing by 108:
p^3 = 1/108
Taking the cube root:
p = 1/6
Now, we can substitute the value of p back into the probability distribution to find the individual probabilities:
1. Selecting 0 non-relative candidates: (3(1/6))^3 = 1/216
2. Selecting 1 non-relative candidate: 3(3(1/6))^2 * (1/6) = 1/12
3. Selecting 2 non-relative candidates: 3(3(1/6)) * (3(1/6))^2 = 1/12
4. Selecting 3 non-relative candidates: (3(1/6)) * (3(1/6))^2 = 1/48
Therefore, the probability distribution for the selection of persons other than the chairman's relatives is as follows:
0 non-relative candidates: 1/216
1 non-relative candidate: 1/12
2 non-relative candidates: 1/12
3 non-relative candidates: 1/48
To find the probability distribution for the selection of persons other than the chairman's relatives, we can use the concept of probability.
Let's denote the probability of selecting a non-relative for a job as P(N) and the probability of selecting a relative as P(R). Given that the chairman selects his relatives 3 times as likely as others, we can express this relationship as:
P(R) = 3P(N)
Since there are only two possibilities (either selecting a relative or selecting a non-relative), the sum of these probabilities should be 1:
P(R) + P(N) = 1
Substituting the relationship between P(R) and P(N):
3P(N) + P(N) = 1
4P(N) = 1
P(N) = 1/4
So, the probability of selecting a non-relative (persons other than the chairman's relatives) is 1/4.
Now, to find the probability distribution for the selection of persons other than their relatives for the 3 job positions, we can use the concept of independent events. Since the events are independent, the probability of selecting a non-relative for each of the 3 job positions would be the same, i.e., 1/4.
Therefore, the probability distribution for the selection of persons other than the chairman's relatives for the 3 job positions is as follows:
P(N1) = P(N2) = P(N3) = 1/4
Note: This assumes that the chairman selects only one person for each job position.