Probability
Conditional Probability
Grade 12
Question:
<p>In a game show "Kaun Banega Dus Crore Pati" The host Mr. Kabir Khan gave the guest Mr Rajesh a choice of three doors: Behind one door is a new shining car; behind the others, nothing. Mr. Rajesh pick a door, say No. 1, and the host, Mr. Kabir Khan who knows what's behind the doors, opens another door, say No. 3, which has nothing. He then says to Mr Rajesh, "Do you want to pick door No. 2?" What he should do now to win the car?</p>
<p>(a) \(2/3\)</p>
<p>(b) \(1/3\)</p>
<p>(c) \(1\)</p>
<p>(d) \(0\)</p>
Step-by-Step Solution
Key Concept: This is the Monty Hall Problem: the host's action of opening a door with nothing behind it concentrates the probability of the car being behind the unopened door (not chosen initially) to 2/3, while your original choice remains 1/3. Switching doubles your winning probability.
<p><strong>Step 1:</strong> Initially, P(car behind door 1) = 1/3 and P(car behind doors 2 or 3) = 2/3.</p><p><strong>Step 2:</strong> The host (who knows the contents) deliberately opens door 3 showing nothing. This action does NOT change the probability that door 1 has the car—it remains 1/3. However, the entire 2/3 probability that the car was among doors 2 and 3 now concentrates on door 2 alone.</p><p><strong>Step 3:</strong> After the host opens door 3: P(car behind door 1) = 1/3 and P(car behind door 2) = 2/3.</p><p><strong>Step 4:</strong> Since P(car behind door 2) = 2/3 > P(car behind door 1) = 1/3, Mr. Rajesh should <strong>switch to door 2</strong>.</p><p>∴ Answer: A (Switch to door No. 2)</p>
Correct Answer: A