GATE CS 2016 Set 2 Q05 — Conditional Probability and Bayes Theorem

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

$$P(T_1) = P(T_2) = 0.5$$

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

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

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

Using the Law of Total Probability, the probability that a randomly chosen bulb lasts more than $100$ hours is:

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

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