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Probabilistic thinking – part 2
Nur Aini Masruroh
Application of conditional probability
Direct conditioning: Relevance of smoking to lung cancer Suppose:
S: A person is a heavy smoker which is defined as having smoked at least two packs of cigarettes per day for a period of at least 10 years during a lifetime
L: A person has lung cancer according to standard medical definition
A doctor not associated with lung cancer treatment assigned the following probabilities:
Relevance of smoking to lung cancer (cont’d)
A lung cancer specialist remarked: “The probability p(L1|S1, ξ) = 0.1 is too low”
When asked to explain why, he said:
“Because in all these years as a lung cancer specialist, whenever I visited my lung cancer ward, it is always full of smokers.”
What’s wrong with the above statement? The answer can be found by flipping the tree:
Relevance of smoking to lung cancer (cont’d)
What the specialist referred to as “high” is actually the probability of a person being a smoker given that he has lung cancer, i.e., p(S1|L1, ξ) = 0.769 is exactly what he was referring to.
He has confused p(S1|L1, ξ) with p(L1|S1, ξ)
Notice that p(L1|S1, ξ) << p(S1|L1, ξ)
Hence even highly a trained professional can fall victim to wrong reasoning
Let’s Make a Deal Game Show
Rules: Consider the TV game show where the contestant is shown o stage
three boxes, one of which contains a valuable prize; the other two are empty
The rules of the game are that the contestant first chooses one of the boxes. Then, the game show host who knows the location of the prize opens one of the remaining two boxes, making sure to open an empty one.
the contestant then gets to decide if he wants to stick with his initial selection or switch to the remaining unopened box.
If the prize is in the box that he finally chooses, he wins the prize
Question:
If at the start of the game the contestant chose box A and the host open box B, should the contestant keep choosing box A or swicth to box C?
Updating probabilities based on new evidence or information
Recall: p(A|ξ) is probability of event A based on our subjective assessment
of the likelihood of event using any information we have prior probability
If new information E has arrived, then the probability of A is updated using Bayes’ Theorem:
where p(E|A) is called the likelihood function for the evidence E and
)(
)|()()|(
Ep
AEpApEAp
j
jj AEpApEp )|()()(
Example: weather forecast
Suppose, the prior probability that it will rain tonight (R1) is 0.6 and it will not rain (R2) with probability 0.4
Suppose we are using information from the weather forecast whose performance is as follows
If the weather station announces that it will rain tonight, what probability should you assign to the outcome it will indeed rain tonight?
Weather forecast (cont’d)
Try to use Bayes’ theorem!
Another example
In the city, there are only two taxicab companies, the Blue and the Green. The Blue company operates 90% of all cabs in the city and the Green company operates the rest. One dark evening, a pedestrian is killed by a hit-and-run taxicab.
There is one witness to the accident. In court, the witness’ ability to distinguish cab colors in the dark is questioned so he is tested under conditions similar to those in which the accident occurred. If he is shown a green cab, he says it is green 80% of the time and blue 20% of the time. If he is shown a blue cab, he says it is blue 80% of the time and green 20% of the time.
The judge believes that the test accurately represents the witness’ performance at the time of the accident.
Construct the probability tree representing the judge’s state of information!
If the witness says “ The cab involved in the accident was green.” What probability should the judge assign to the cab involved in the accident being green?