GO Classes DPP | GATE DA | Probability | Conditional Probability

Answer 1

Let’s suppose you choose Door $1$, since the same analysis applies whichever door you chose.

$\textbf{Strategy 1 (Stick):}$ With probability $1/3$ you chose the car initially. Monty Hall shows you one of the other doors, but that doesn’t change your probability of winning. The probability of winning by sticking is $1/3$.

$\textbf{Strategy 2 (Switch):}$

• $\textbf{Case 1:}$ The car is behind Door $1$, which happens with probability $1/3$. Monty Hall shows you one of the other doors, say Door $2$. There will be a goat, so you switch to Door $3$, and lose. The same argument applies if he shows you Door $3$.
   
• $\textbf{Case 2:}$ The car is behind Door $2$. He will show you Door $3$, since he doesn’t want to give away the car. You switch to Door $2$ and win. This happens with probability $1/3$.
   
• $\textbf{Case 3:}$ The car is behind Door $3$. He will show you Door $2$. You switch to Door $3$ and win. This happens with probability $1/3$.
   

So, you win with probability $1/3+1/3=2/3$ and lose with probability $1/3$.

Thus, strategy $2$ (switching) is much superior.

Answer 2

added

Answer 3

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Answer 4

Classic trap—intuition says “now 2 doors ⇒ 1/2 each”, but Monty is not random. He reveals information means Monty’s action depends on hidden information (where the car is)—he is not opening a door randomly.

1) If Monty were random (no knowledge)

• He might accidentally open the car door
• If he opens a goat, then yes—now it’s 50–50

👉 Here intuition “2 doors ⇒ 1/2” would be correct.

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2) In the real problem (Monty knows)

• He always avoids the car
• He always shows a goat
• His choice is constrained, not random

👉 So his action gives information.

We First pick	Where is car?	What Monty opens	If switch	Result
Door 1	Door 1	Goat door (2,3)	Switch → Goat	Lose
Door 1	Door 2	Goat door(1(chosed by us) ,3 (this is left to open)) -> (3)	Switch→ Car	Win
Door 1	Door 3	Goat door(1(chosed by us) ,2 (this is left to open)) -> (2)	Switch → Car	Win

• If stick → win only when initial pick correct → 1/3
• If switch → win in 2 cases → 2/3

Hence Switch. Probability of winning = 2/3​​ (Option B)