(All Steps) You are a contestant in the usual Monty Hall game show with five closed doors. Behind one of the doors is the car of your dreams. Behind the
Question: You are a contestant in the usual Monty Hall game show with five closed doors. Behind one of the doors is the car of your dreams. Behind the other four doors are smelly goats. You first choose a door, after which the host opens a different door (i.e., not the one you chose). The door the host opens always has a goat behind it. Then the host asks if you would like to switch, i.e., choose a door other than the one you chose the first time. Calculate the probability of winning, assuming that you always switch. Is this higher than the probability of winning, assuming that you ignore the host and stay with your original guess? You may assume that when faced with multiple doors to choose from, you choose each door with equal probability.
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![[See Solution] Let X be a random](/images/solutions/MC-solution-library-53656.jpg "[See Solution] Let X be a random variable with pdf f(x)=3 / 14 √x+1")
[See Solution] Let X be a random variable with pdf f(x)=3 / 14 √x+1
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![[Solution] Using the uniform distribution on $[0,1]$,](/images/solutions/MC-solution-library-53659.jpg "[Solution] Using the uniform distribution on $[0,1]$, we choose")
[Solution] Using the uniform distribution on $[0,1]$, we choose
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