Conditional Probability & Conditional Expectations Compute - - PowerPoint PPT Presentation

Conditional Probability & Conditional Expectations

Compute Probabilities by Conditioning

IE 502: Probabilistic Models Jayendran Venkateswaran IE & OR

IE502: Probabilistic Models IEOR @ IITBombay

Gambler’s Chances

• A gambler has Rs.300 in his pocket when he

walks into a casino. He plays a game in which the probability he gains Rs.100 is 1/3, the probability that he gains Rs.200 is 1/6, and the probability that he loses Rs.100 is 1/2. He decides ahead of time that he will stop playing as soon as he has enough money to buy a Rs.500 toy for his daughter. Of course, he also must stop playing if he goes broke!

– What is the probability that he goes broke? …buys toy? – What is the expected number of games he plays before he leaves the casino?

IE502: Probabilistic Models IEOR @ IITBombay

Welcome to the game show! a.k.a Monty Hall

• We have two goats

and a car.

• They are placed

randomly behind closed doors.

• You, the contestant,

need to pick a door, & win what’s inside!

• But with a twist..

IE502: Probabilistic Models IEOR @ IITBombay

Game show a.k.a Monty Hall

• You are given a

choice of 3 doors. Behind one door is a car & behind the

• ther two, goats
• What’s the probability
• f winning a car?

IE502: Probabilistic Models IEOR @ IITBombay

Game show a.k.a Monty Hall

• You are given a

choice of 3 doors. Behind one door is a car & behind the

• ther two, goats
• What’s the probability
• f winning a car?

P{Car} = 1/3 P{Car} = 1/3 P{Car} = 1/3

IE502: Probabilistic Models IEOR @ IITBombay

Game show a.k.a Monty Hall

• You choose a door

Say, door 3 [door 3 remains closed]

• The twist…

IE502: Probabilistic Models IEOR @ IITBombay

Game show a.k.a Monty Hall

• You choose a door

say, door 3 [door 3 remains closed]

• The twist…
• I open one of the

remaining doors [either 1 or 2] to reveal a goat

IE502: Probabilistic Models IEOR @ IITBombay

Game show a.k.a Monty Hall

Now, “Do you wish to stay with your original choice (door3) or switch (to door1)?”

• You choose a door

say, door 3 [door 3 remains closed]

• The twist…
• I open one of the

remaining doors [either 1 or 2] to reveal a goat

IE502: Probabilistic Models IEOR @ IITBombay

Game show a.k.a Monty Hall

Now, “Do you wish to stay with your original choice (door3) or switch (to door1)?”

• You choose a door

say, door 3 [door 3 remains closed]

• The twist…
• I open one of the

remaining doors [either 1 or 2] to reveal a goat

P{Car} = ? P{Car} = 0 P{Car} = ?