I was watching CNN at work the morning of September 11th, 2001 once the first plane hit the World Trade Center. On my way into work, I would have assigned almost no probability to terrorists crashing planes into buildings that day. The first plane hit the World Trade Center and we thought it might be an accident, but once the second plane hit, I immediately headed home to be with my family, because I was sure we were under a terror attack.

Use Bayes Theorem to replicate this result.

Prior: Use an initial estimate (prior) of the possibility of a terror attack at a very low 0.005%

Event: The first plane hits the North Tower of the World Trade Center at 8:46am EST

Problem 1: Likelihood of an accident: In the previous 25,000 days of aviation over Manhattan before September 11th, there had been two airplane accidents, one involving the Empire State Building in 1945 and another at 40 Wall Street in 1946. Calculate the likelihood of an accident based on these numbers, assuming an accident was equally likely each day.

Problem 2: Let's assume that if a terrorist attack on Manhattan is underway, the probability of seeing evidence like what we are seeing is very high, say 100%. Where should we use this information?

a) P(E|H)

b) P(H)

c) P(E|-H)

d) P(H|E)

e) P(-H)

Problem 3: We are trying to calculate the posterior probability, the likelihood of a terror attack given the first plane hitting the World Trade Center. What is the posterior probability?

a) P(E|H)

b) P(H)

c) P(E|-H)

d) P(H|E)

e) P(-H)

Problem 4: Calculate the posterior probability, the likelihood of a terror attack, given the first plane hitting the World Trade Center.

Event: The second plane hits the South Tower of the World Trade Center at 9:03am EST

Problem 5: Using your revised estimate of the probability of a terror attack in Problem 3 as the prior, calculate the posterior probability of a terror attack, given the second plane hitting the World Trade Center.