Prob - difficulty following exemple

Thread starter quasar987
Start date Oct 9, 2006
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The discussion centers on calculating the probability of achieving n consecutive successes before m consecutive failures in Bernoulli processes with respective probabilities p and q. The solution involves conditioning on the first process and defining events E, F, G, and H. The confusion arises from the claim that P(E|FG^c) equals P(E|F^c), which is contested by the participants, as G^c does not imply all failures, contradicting the probability q^{n-1} for all failures. The participants agree on the correctness of P(E|FG) and P(G|F) but challenge the validity of the assumptions made in the solution.

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Oct 9, 2006

quasar987

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We perform Bernoulli processes of respective probabilities p and q. What is the probability of getting n consecutive successes before getting m consecutive failures?

Sol: We define the following events:

E: getting n consecutive successes before getting m consecutive failures.

F: The first process is a success.

G: The n-1 processes following the first are successes.

H: The m-1 processes following the first are failures.

We condition on the first process:

[tex]P(E)=P(E|F)P(F)+P(E|F^c)P(F^c)[/tex]

We condition P(E|F) on the event G:

[tex]P(E|F)=P(E|FG)P(G|F)+P(E|FG^c)P(G^c|F)[/tex]

The solution then says that P(E|FG)=1 and P(G|F)=[itex]p^{n-1}[/itex]. On that I agree. But it also says that [itex]P(E|FG^c)=P(E|F^c)[/itex] and [itex]P(G^c|F)=q^{n-1}[/itex]. I can't make any sense of the first one, but the second is obviously false, because [itex]G^c[/itex] means "the n-1 processes following the first are not all successes", while [itex]q^{n-1}[/itex] is the probability for the event "the n-1 processes following the first are all failures".

But "not all sucesses" does not imply "all failures". Am I crazy?