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