Transcript Probability - Berkeley Cosmology Group

Monty Hall Problem
Presented By:
Netsay Ramos
Lorrain Manasan
Refilwe Gqajela
Kevin Gil
Overview:The Monty Hall Problem
Suppose you're on a game show, and you're given the
choice of three doors:
• Behind one door is a car; behind the other two are goats.
• You pick a door, say No. 1, and the host, who knows what's
behind the doors, opens another door, say No. 3, which has
a goat.
• He then says to you, "Do you want to pick door No. 2?"
• Is it to your advantage to switch your choice?
Card Experiment
Hypothesis: About 50% of the time, switching will make you
win, and about 50% of the time, staying will make you win.
Number of Trials: 260
Switching won 174 times = ~ 66.9%= ~2/3
Staying won 86 times = ~ 33.0% = ~1/3
Clearly not in line with our hypothesis!
Law of Large Numbers
Law of Large Numbers: a valid way to test probability is to
perform the same experiment a large number of times.
Example: Flipping a coin.
• We know that there is a 50/50
chance of landing heads or tails
• Flip a coin 100,000 times
• The more you flip the coins, the
close your get to 50%
Interpretations of Results
Using law of large numbers, the probability that switching
makes you win is about 2/3, and the probability that staying
makes you win is about 1/3!
o We conducted the experiment several times to get as
close to what the expected probability of each scenario
would be.
o
Our numbers came close but not exact, this is because
the probability is idealistic.
o
Ex. Flipping a coin- heads vs. tails is 1/2, but if you do it
500 times, you won't get exactly 250 vs. 250 but very
close to it
What is Bayes Theorem?
Bayes Theorem is the correlation between P(A|B) and P(B|A).
• P(A|B) or Probability of A given B, means that with previous
knowledge of B happening, the probability that A would
would in turn happen.
• P(A|B) = P(A) P(B|A)
P(B)
Bayes Theorem for Staying
A is the event that you initially chose correctly (the car)
B is the event that after you pick your initial choice, Monty
opens a door to reveal the goat.
We are trying to find the probability that you initially picked
right, providing that he opened the door to reveal a goat and
you chose to stay. This is P(A|B).
Probability of B|A=1
Probability of A=1/3
Probability of B=1
Recall Bayes Theorem:
P(A|B)=1X1/3 = 1/3
1
In this situation you are less likely to win if you choose to stay.
Thank you for your time!
Any questions?
References
1. <http://www.grandillusions.com/images/articles/articles/monty_hall/mainimage.
jpg>
2. <http://coins.about.com/od/coinsglossary/ss/coinanatomy.ht
m>
3. Other references available upon request