Prob - difficulty following exemple

[quasar987]

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:

$$P(E)=P(E|F)P(F)+P(E|F^c)P(F^c)$$

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

$$P(E|F)=P(E|FG)P(G|F)+P(E|FG^c)P(G^c|F)$$

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

 

[lippyka]

No, you are not crazy. The solution is incorrect and should not state that P(E|FG^c)=P(E|F^c) and P(G^c|F)=q^{n-1}. Instead, it should be P(E|FG^c)=P(E|F^c) and P(G^c|F)=1-p^{n-1}.