<p>In the Monty Hall Problem: Mr. Rajesh is on a game show and is given the choice of three doors. Behind one door is a car and behind the others are empty. Mr. Rajesh picks door No. 1. The host (who knows what's behind the doors) opens door No. 2 and reveals it is empty. The host then asks Mr. Rajesh if he wants to switch to door No. 3. What is the probability of winning if Mr. Rajesh switches to door No. 2?</p>
Step-by-Step Solution
Key Concept: Switching doors changes your probability from 1/3 to 2/3 because the host's action of revealing an empty door non-randomly (knowing what's behind) updates the conditional probability. The host eliminates a door strategically, concentrating the remaining 2/3 probability on the unopened door you didn't choose.
<p><strong>Step 1: Initial Setup</strong><br/>Three doors: 1 car, 2 empty. Mr. Rajesh picks door 1.<br/>P(car behind door 1) = 1/3<br/>P(car behind door 2 or 3) = 2/3</p><p><strong>Step 2: Host Opens Door 2</strong><br/>Host knows what's behind each door and deliberately opens an empty door (door 2). This is KEY—the host doesn't open randomly.</p><p><strong>Step 3: Conditional Probability After Revelation</strong><br/>The host opening door 2 (empty) tells us information. Given that door 1 was initially chosen with probability 1/3 of having the car, the conditional probability that the car is behind door 3 is:<br/>P(car behind door 3 | door 2 opened empty) = P(car behind doors 2 or 3) = 2/3<br/><br/>This is because if the car was among doors {2,3} (probability 2/3), the host must open door 2, guaranteeing the car is in door 3.</p><p><strong>Step 4: Switching Strategy</strong><br/>If Mr. Rajesh switches from door 1 to door 3:<br/>P(winning by switching) = 2/3</p><p><strong>Note:</strong> The question text says 'switch to door No. 3' but then asks about 'switching to door No. 2'—there appears to be a typo in the problem statement. Assuming he switches to door 3 (the remaining unopened door), the probability is 2/3.</p><p>∴ Answer: A (2/3)</p>
Correct Answer: A