Probability of Selecting 2 Men from a Group
This section details the calculation for the probability of selecting exactly 2 men when 3 people are chosen randomly from a mixed group.
Group Composition Overview
The total group comprises:
- Number of Men: 4
- Number of Women: 3
- Number of Boys: 2
- Total Persons: $4 + 3 + 2 = 9$
Total Selections Calculation
The total number of ways to select 3 persons from 9 is found using combinations ($C(n, k) = \frac{n!}{k!(n-k)!}$):
Total ways = $C(9, 3) = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$
Favorable Selections Calculation
Favorable outcomes require selecting exactly 2 men and 1 other person (non-man).
- Ways to choose 2 men out of 4:
$C(4, 2) = \frac{4!}{2!2!} = \frac{4 \times 3}{2 \times 1} = 6$
- Ways to choose 1 person from the remaining 5 (3 women + 2 boys):
$C(5, 1) = \frac{5!}{1!4!} = 5$
- Total favorable ways = (Ways to choose men) $\times$ (Ways to choose other)
$6 \times 5 = 30$
Probability Calculation Step
Probability is calculated as the ratio of favorable outcomes to total possible outcomes:
Probability = $\frac{\text{Favorable Ways}}{\text{Total Ways}} = \frac{30}{84}$
Simplifying the fraction gives:
$\frac{30 \div 6}{84 \div 6} = \frac{5}{14}$
Final Probability Result
The probability of selecting exactly 2 men is $\frac{5}{14}$.
Similar Questions
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Important Questions from Probability
Two distinct natural numbers from 1 to 9 are picked at random. What is the probability that their product has 1 in its unit place?
Two dice are thrown. What is the probability that difference of numbers on them is 2 or 3 ?
Suppose that there is a chance for a newly constructed building to collapse, whether the design is faulty or not. The chance that the design is faulty is 10%. The chance that the building collapses is 95% if the design is faulty, otherwise it is 45%. If it is seen that the building has collapsed, then what is the probability that it is due to faulty design?
What is the probability that boys and girls sit alternatively?
What is the probability that P and Q take the two end positions?
