GATE CS 2016 Set 2 Q.15 (CS Q.5): Probability and Statistics

Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than 100 hours given that it is of Type 1 is 0.7, and given that it is of Type 2 is 0.4. The probability that an LED bulb chosen uniformly at random lasts more than 100 hours is ________.

Your answer

Official answer and solution

Official answer: 0.55
From: Official GATE 2016 answer key

Solution

Use the Law of Total Probability over the two bulb types.

Let $T_1$ and $T_2$ denote the events that a randomly selected LED bulb is of Type 1 and Type 2, respectively. Since the shop has an equal number of bulbs of both types:

P(T_1) = P(T_2) = 0.5

Let $L$ denote the event that a bulb lasts more than 100 hours. The given conditional probabilities are:

$P(L \mid T_1) = 0.7$
$P(L \mid T_2) = 0.4$

Applying the Law of Total Probability:

P(L) = P(T_1) P(L \mid T_1) + P(T_2) P(L \mid T_2)

P(L) = (0.5)(0.7) + (0.5)(0.4) = 0.35 + 0.20 = 0.55

Therefore, the probability that a bulb chosen uniformly at random lasts more than 100 hours is 0.55.

Final answer

0.55

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2016 Computer Science and Information Technology (CS) Session 2
Question: 15 of 65, printed as CS Q.5
Section: Computer Science and Information Technology

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