GATE Data Science and Artificial Intelligence 2024 | Sample Paper | Question: 45

Question

A class contains $60 \%$ students who are incapable of changing their opinions about anything, and $40 \%$ of students are changing their minds at random, with probability $0.3$, between subsequent votes on the same issue. Then, the probability of a student randomly chosen voted twice in the same way is ____________.

Answer 1

Stubborn Students (60%):

These students always vote the same way, so the probability of them voting twice in the same way is simply 1 (100%).

2. Changing-Mind Students (40%):

These students can vote either way on the second vote, so the probability depends on the chance of them keeping their original vote:

• Probability of keeping original vote: (1 - 2 * probability of changing mind) + probability of changing mind^2
  • This accounts for three possibilities: keeping the original vote (1 - 2 * 0.3), switching then switching back (2 * 0.3 * 0.7), and never changing (0.3^2
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  • Now, we need to combine the probabilities for both types of students weighted by their proportions in the class:
• Probability of randomly chosen student voting same way twice = (proportion of stubborn students * probability they vote same way) + (proportion of changing-mind students * probability they keep original vote)
• This translates to: 0.6 * 1 + 0.4 * (1 - 2 * 0.3 + 0.3^2) ≈ 0.7960

Comments

I didn’t understand. Can you provide any reference to this concept?

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Wrong. Correct answer will be: 0.6 * 1 + 0.4 * 0.7 = 0.88

0.6 is the probability of choosing students with fixed opinions.

For, the rest the chances of changing opinions is 0.3 between subsequent votes (see in the question). That means the chances that the 40% students will not change their opinions between first and second vote is 0.7.

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Ranker h ye. Solving questions and posting them here for the sample paper. Thanks!