#### haushofer

Science Advisor

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Religious comments deleted throughout the thread

Dear all,

Imagine you're participating in a lottery, with a probability of 1 in 10 million to win. Then your newspaper is giving the outcomes of the lottery, and it turns out: you've won! But being a sceptic, you doubt the reliability of the newspaper. Say, historically this newspaper publishes the wrong lottery outcome 1 in every 50 of their lottery-reports (just say; it would be pretty bad!). How would you calculate the probability that you've actually won?

So, given:

[tex]

P(\text{won}) = 10^{-7}, P(\text{article contains error}) = 0.02, P(\text{article contains no error}) = 0.98

[/tex]

what's

[tex]

P(won | article \ contains \ no \ error)

[/tex]

? The book argues:

"Why is it reasonable to believe you've won? Because although the probability of you winning is extremely small, the probability of the newspaper giving you the correct outcome is much bigger."

Somehow, I feel a bit itchy about this argument (as about a lot of other arguments in the book) and I would be tempted to use Bayes' theorem.

I know about the well-known examples of testing for illness, where a very reliable test can still give a lot of false outcomes if the illness is a priori very rare; a particular example is e.g. an HIV-test; if a priori one out of ten thousand people is infected, and the reliability of the test is such that it gives a correct result for 99% of infected people and a correct result for 99.9% of not-infected people, then Bayes theorem gives with

[tex]

P(infected) = \frac{1}{10000}, \ \ \ P(positive| infected) = 0.99 , \ \ \ P(negative| infected) = 0.01 \nonumber\\

P(negative | not \ infected) = 0.999 , \ \ \ P(positive | not \ infected) = 0.001

[/tex]

the outcome

[tex]

P(infected | positive) = 0.09 = 9 \%

[/tex]

This makes sense: out of 10.000 people we expect one infected person, while the test also gives approximately 10 false positives, making the propability ##P(infected | positive)## approximately 1 on out 11. Bayes theorem indicates that ##P(infected | positive)## is proportional to ##P(infected)##, and intuitively I'd say this is also true for our lottery. But then again, the lottery case is different, because every participant receives the same outcome of the newspaper. So my questions are basically:

[*] How does the illness example relate to the lottery example?

[*] What's the probability of me having actually won the lottery given the positive newspaper outcome, ##P(won | article \ contains \ no \ error)##? Can I trust my newspaper given the highly improbable event of winning?

[*] What do you think about the book's argument ""It's reasonable to trust the newspaper because although the probability of you winning is extremely small, the probability of the newspaper giving you the correct outcome is much bigger." ?

[*] Can the lottery example indeed be used to motivate the existance of highly impropable events? I.e. is the probability for the occurence of highly improbable events independent of the a priori probability of these events (whatever that number may be), contrary to the illness example?

Thanks in advance!

Imagine you're participating in a lottery, with a probability of 1 in 10 million to win. Then your newspaper is giving the outcomes of the lottery, and it turns out: you've won! But being a sceptic, you doubt the reliability of the newspaper. Say, historically this newspaper publishes the wrong lottery outcome 1 in every 50 of their lottery-reports (just say; it would be pretty bad!). How would you calculate the probability that you've actually won?

So, given:

[tex]

P(\text{won}) = 10^{-7}, P(\text{article contains error}) = 0.02, P(\text{article contains no error}) = 0.98

[/tex]

what's

[tex]

P(won | article \ contains \ no \ error)

[/tex]

? The book argues:

"Why is it reasonable to believe you've won? Because although the probability of you winning is extremely small, the probability of the newspaper giving you the correct outcome is much bigger."

Somehow, I feel a bit itchy about this argument (as about a lot of other arguments in the book) and I would be tempted to use Bayes' theorem.

I know about the well-known examples of testing for illness, where a very reliable test can still give a lot of false outcomes if the illness is a priori very rare; a particular example is e.g. an HIV-test; if a priori one out of ten thousand people is infected, and the reliability of the test is such that it gives a correct result for 99% of infected people and a correct result for 99.9% of not-infected people, then Bayes theorem gives with

[tex]

P(infected) = \frac{1}{10000}, \ \ \ P(positive| infected) = 0.99 , \ \ \ P(negative| infected) = 0.01 \nonumber\\

P(negative | not \ infected) = 0.999 , \ \ \ P(positive | not \ infected) = 0.001

[/tex]

the outcome

[tex]

P(infected | positive) = 0.09 = 9 \%

[/tex]

This makes sense: out of 10.000 people we expect one infected person, while the test also gives approximately 10 false positives, making the propability ##P(infected | positive)## approximately 1 on out 11. Bayes theorem indicates that ##P(infected | positive)## is proportional to ##P(infected)##, and intuitively I'd say this is also true for our lottery. But then again, the lottery case is different, because every participant receives the same outcome of the newspaper. So my questions are basically:

[*] How does the illness example relate to the lottery example?

[*] What's the probability of me having actually won the lottery given the positive newspaper outcome, ##P(won | article \ contains \ no \ error)##? Can I trust my newspaper given the highly improbable event of winning?

[*] What do you think about the book's argument ""It's reasonable to trust the newspaper because although the probability of you winning is extremely small, the probability of the newspaper giving you the correct outcome is much bigger." ?

[*] Can the lottery example indeed be used to motivate the existance of highly impropable events? I.e. is the probability for the occurence of highly improbable events independent of the a priori probability of these events (whatever that number may be), contrary to the illness example?

Thanks in advance!

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