Question

In the three-door Monty Hall problem, there are two stages to the decision, the initial pick followed by the decision to stick with it or switch to the only other remaining alternative after the host has shown an incorrect door. An extension of the basic problem to multiple stages goes as follow. Suppose there are four doors, one of which is a winner. The host says: You point to one of the doors, and then I will open one of the other non-winners. Then you decide whether to stick with your original pick or switch to one of the remaining doors. Then I will open another (other than the current pick) non-winner. You will then make your final decision by sticking with the door picked on the previous decision or by switching to the only other remaining door. (a) How many possible strategies are there? (b) For each of the possible strategies, calculate the probability of winning. Proceed by systematically defining a probability model as we did in class. What is the best strategy?

Answer 1

In a Monty Hall Problem with four doors, in which only one door has the price and the rest are goats.

(a) There are four different possible strategies.

There were only two solutions in the original Monty Hall problem the sticking strategy where you'd stick with your original pick after Monty opens the first door, or the switching strategy where you'd change your original pick once Monty opens the first door.
Now, there are two stages where a decision is required, and there are two choices at each stage. These are

• Either you stick with your current door, or
• you can switch to another door (stage 1 has two possible doors to choose from, choose one of them randomly)

This gives you 2*2 = 4 different strategies as shown in the table below.

Stage	1	2	3	Probability of winning
Strategy 1	Pick	Stick	Stick	??
Strategy 2	Pick	Switch	Stick	??
Strategy 3	Pick	Stick	Switch	??
Strategy 4	Pick	Switch	Switch	??

(b) Now let's complete the table by calculating the Probability of winning for each strategy

I will use the following notation

• WWW = to denote picking the Winning door in each of the 3 times
• LLL = to denote picking the Losing door in each of the 3 times

So, just to be clear, WLW will correspond to Win Lose Win

Note that we only Win, when we choose the Winning door in the last time.

Strategy 1: Stick in both

Here we can have the following cases:

• Pick the Winning door first time and stick to it, so WWW
  Probability(WWW) = Pr(picking the wining door out of 4 doors in first time) = 1/4 => YOU WIN
• Pick any of the three wrong doors in the first time and stick to it
  Probability(WWW) = Pr(picking 3 wrong door out of 4 doors in first time) = 3/4 => YOU LOSE

Since you only won in the 1st case i.e. WWW, so P(Winning | Strategy 1) = 1/4 = 0.25

Strategy 2: Switch and Stick

Here we can have the following cases:

• Pick the Winning door first time, Switch to Losing door, Stick to it, so WLL
  P(WLL) = 1/4 = 0.25 => YOU LOSE
• Pick the Losing door first time, Switch to Another Losing door, Stick to it, so LLL
  P(LLL) = 3/4*1/2 = 0.375 => YOU LOSE
• Pick the Losing door first time, Switch to Winning door, Stick to it, so LWW
  P(LWW) = 3/4*1/2 = 0.375 => YOU WIN

​​​​​​​Since you only won in the 3rd case i.e. LWW, so P(Winning | Strategy 2) = 0.375

Strategy 3: Stick and Switch

Here we can have the following cases:

• Pick the Winning door first time, Stick to it, and Switch to losing door, so WWL
  P(WWL) = 1/4 = 0.25 => YOU LOSE
• Pick the Losing door first time, Stick to Losing door, and Switch to Winning door, so LLW
  P(LLW) = 3/4 = 0.75 => YOU WIN

​​​​​​​Since you only won in the 2nd case i.e. LLW, so P(Winning | Strategy 3) = 0.75

Strategy 4: Switch and Switch

Here we can have the following cases:

• Pick the Winning door first time, Switch to Losing door, and Switch to Winning door, so WLW
  P(WLW) = 1/4 = 0.25 => YOU WIN
• Pick the Losing door first time, Switch to Winning door, and Switch to Losing door, so LWL
  P(LWL) = 3/4*1/2 = 0.375 => YOU LOSE
• Pick the Losing door first time, Switch to Losing door, and Switch to Winning door, so LLW
  P(LLW) = 3/4*1/2 = 0.375 => YOU WIN

​​​​​​​Since you won in the 1st and 3rd case i.e. WLW and LLW, so P(Winning | Strategy 4) = 0.25 + 0.375 = 0.625

So now finally we can complete our table as follows:

Stage	1	2	3	Probability of winning
Strategy 1	Pick	Stick	Stick	0.25
Strategy 2	Pick	Switch	Stick	0.375
Strategy 3	Pick	Stick	Switch	0.75
Strategy 4	Pick	Switch	Switch	0.625

​​​​​So Strategy 3 i.e Stick and Switch is the best strategy for winning.

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