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Difficulty: HardProbability of Independent, Dependent, and Mutually Exclusive Events

A committee of 33 members is to be selected at random without replacement from a group of 55 data scientists and 55 software engineers. What is the conditional probability that at least two data scientists are selected in total, given that the first person selected is a data scientist?

  1. A
    518\frac{5}{18}
  2. B
    12\frac{1}{2}
  3. C
    59\frac{5}{9}
  4. 1318\frac{13}{18}Answer
  5. E
    34\frac{3}{4}

Answer

The conditional probability is 1318\frac{13}{18}.
The conditional probability is calculated by adjusting the sample space after the first draw. With 1 data scientist already selected, there are 4 data scientists and 5 software engineers left (9 total). Selecting at least 1 more data scientist in the next 2 draws has a probability complementary to selecting 0 more data scientists. The probability of selecting 0 additional data scientists is (52)(92)=1036=518\frac{\binom{5}{2}}{\binom{9}{2}} = \frac{10}{36} = \frac{5}{18}. Therefore, the required conditional probability is 1−518=13181 - \frac{5}{18} = \frac{13}{18}.

Step-by-Step Solution

1
Determine the remaining pool of candidates after the first selection.
Since 1 data scientist is selected first, 4 data scientists and 5 software engineers remain (total of 9 candidates). Two more candidates must be selected.
The conditional statement fixes the first selection, changing the sample space for the remaining 2 selections.
2
Identify the condition required for the target event to occur.
Since 1 data scientist is already selected, having at least two data scientists in total means selecting at least 1 additional data scientist in the remaining 2 draws.
Total data scientists = 1 (first draw) + (number of data scientists in next 2 draws).
3
Calculate the complementary probability (selecting 0 additional data scientists).
The number of ways to pick 2 software engineers from 5 is (52)=10\binom{5}{2} = 10. The total ways to pick 2 people from 9 is (92)=36\binom{9}{2} = 36. The probability of 0 additional data scientists is 1036=518\frac{10}{36} = \frac{5}{18}.
Selecting 0 additional data scientists is equivalent to selecting 2 software engineers from the remaining group.
4
Subtract the complementary probability from 1.
1−518=13181 - \frac{5}{18} = \frac{13}{18}.
The sum of the probability of an event and its complement equals 1.

Key Concept

Conditional Probability without Replacement
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Topics

Probability of Independent, Dependent, and Mutually Exclusive Events

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