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Difficulty: MediumProbability of Independent and Dependent Events

A crate in a warehouse contains 1515 functional electronic components and 1010 defective components. If two components are selected at random from the crate one after another without replacement, what is the probability that both selected components are functional?

Answer: 0.35

Answer

The probability that both selected components are functional is 0.35.
Because selection is made without replacement, the outcome of the second draw depends on the outcome of the first draw. The probability of selecting a functional component first is 15/25 = 3/5. With 14 functional components left among 24 total components, the probability of selecting a second functional component is 14/24 = 7/12. Multiplying these probabilities gives (3/5) * (7/12) = 21/60 = 0.35.

Step-by-Step Solution

1
Calculate the probability of drawing a functional component on the first selection.
P(First Functional) = 15 / 25 = 3/5 = 0.6.
There are 15 functional components in the total pool of 25 components.
2
Calculate the conditional probability of drawing a second functional component given that the first component drawn was functional.
P(Second Functional | First Functional) = 14 / 24 = 7/12.
Because sampling is without replacement, the total count decreases to 24 and the functional count decreases to 14.
3
Multiply the two dependent probabilities to find the overall joint probability.
P(Both Functional) = (3/5) * (7/12) = 21/60 = 0.35.
By the multiplication rule for dependent events, P(A and B) = P(A) * P(B|A).

Key Concept

Probability of Dependent Events (Sampling Without Replacement)
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Topics

Probability of Independent and Dependent Events

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