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Zorluk: OrtaBasic Probability and Counting Methods

A project team of 33 software developers is to be randomly selected from a pool of 44 frontend developers and 66 backend developers. What is the probability that the selected team will consist of exactly 11 frontend developer and 22 backend developers?

  1. A
    16\frac{1}{6}
  2. B
    310\frac{3}{10}
  3. C
    19120\frac{19}{120}
  4. 12\frac{1}{2}Cevap
  5. E
    325\frac{3}{25}

Cevap

The probability that the team consists of exactly 1 frontend developer and 2 backend developers is 12\frac{1}{2}.
To find the probability, divide the number of ways to select 1 frontend developer and 2 backend developers by the total number of ways to choose any 3 developers from 10. There are (41)=4\binom{4}{1} = 4 ways to select 1 frontend developer and (62)=15\binom{6}{2} = 15 ways to select 2 backend developers. Multiplying these yields 4×15=604 \times 15 = 60 favorable team combinations. The total number of ways to choose 3 developers from 10 is (103)=120\binom{10}{3} = 120. Thus, the probability is 60120=12\frac{60}{120} = \frac{1}{2}.

Adım Adım Çözüm

1
Calculate the total number of possible 3-developer teams from the pool of 10 developers.
(103)=10×9×83×2×1=120\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 total outcomes.
The total number of ways to choose 3 people out of 10 without regard to order is given by combinations.
2
Calculate the number of ways to select 1 frontend developer from 4 and 2 backend developers from 6.
(41)×(62)=4×6×52×1=4×15=60\binom{4}{1} \times \binom{6}{2} = 4 \times \frac{6 \times 5}{2 \times 1} = 4 \times 15 = 60 favorable outcomes.
By the Fundamental Counting Principle, multiply the number of ways to choose the frontend developer by the number of ways to choose the backend developers.
3
Divide the number of favorable outcomes by the total possible outcomes to find the probability.
Probability=60120=12\text{Probability} = \frac{60}{120} = \frac{1}{2}.
Probability is defined as the ratio of favorable outcomes to total outcomes in a uniform sample space.

Anahtar Kavram

Probability of Compound Events using Combinations
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Konular

Basic Probability and Counting Methods

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