GATE 2018 CS – Question 54
Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high (H), medium (M) and low (L). Let $P(H_G)$ denote the probability that Guwahati has high temperature. Similarly, $P(M_G)$ and $P(L_G)$ denotes the probability of Guwahati having medium and low temperatures respectively. Similarly, we use $P(H_D)$, $P(M_D)$ and $P(L_D)$ for Delhi.
The following table gives the conditional probabilities for Delhi's temperature given Guwahati's temperature.
| $H_D$ | $M_D$ | $L_D$ | |
|---|---|---|---|
| $H_G$ | 0.40 | 0.48 | 0.12 |
| $M_G$ | 0.10 | 0.65 | 0.25 |
| $L_G$ | 0.01 | 0.50 | 0.49 |
Consider the first row in the table above. The first entry denotes that if Guwahati has high temperature then the probability of Delhi also having a high temperature is 0.40; i.e., $P(H_D|H_G)=0.40$. Similarly, the next two entries are $P(M_D|H_G)=0.48$ and $P(L_D|H_G)=0.12$. Similarly for the other rows.
If it is known that $P(H_G)=0.2$, $P(M_G)=0.5$, and $P(L_G)=0.3$, then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is ________.
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Show answer and explanation
Correct answer: 0.60 to 0.62
Explanation
By Bayes' theorem, $P(H_G|H_D)=\frac{0.4\times0.2}{0.4\times0.2+0.1\times0.5+0.01\times0.3}=\frac{0.08}{0.133}=0.60\approx0.61$ (the official range is 0.60 to 0.62).