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You repeatedly evaluate a certain discrete R.V. (e.g., discrete triangular) $X_i$ (i.i.d. from one another). What's the probability that two consecutive $a$'s appear before a $b$ appears ($a$ and $b$ are particular realizations that $X_i$ can take).

If it were immediately before a $b$, I assume it would be simply $P(X_i=a)^2$, but I am not restricting it to strictly immediately before $b$.

Would it be correct to use the binomial distribution with $$n= 1/P(X_i = b) - 2$$ The geometric mean. Subtract 2 because I ignore the slot when $b$ occurs and am looking at two consecutive slots to fill, so the first slot doesn't count) and

$$p = P(X_i=a)^2$$ ?

I've seen a much simpler way for this type of question that appears to use some sort of version of the law of total probability (including a recursion term, i.e., the probability being asked for itself), but I wasn't able to see the rationale behind it.

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1 Answer 1

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Let $p_a$ be the probability of an $a$ and $p_b$ be the probability of a $b$.

There are various possibilities for what happens initially:

  • $X_1=b$ with probability $p_b$
  • $X_1=a$ and $X_2=b$ with probability $p_a p_b$
  • $X_1=a$ and $X_2=a$ with probability $p_a^2$
  • $X_1=a$ and $X_2=$ something else with probability $p_a(1-p_a-p_b)$
  • $X_1=$ something else with probability $1-p_a-p_b$

In the first and second case $b$ appears first, while in the third two consecutive $a$s appear first, and in the fourth and fifth you have effectively to repeat the process and carry on.

So overall, two consecutive $a$s appear earlier than any $b$ with probability $$\frac{p_a^2}{p_a^2+ p_a p_b +p_b}.$$

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Thanks for your answer, but I don't quite get the fraction. Is this obtained via the definition of conditional probability? Also, why is $p_aP_b$ included: Doesn't the same logic of " effectively to repeat the process" apply? –  Wuschelbeutel Kartoffelhuhn Oct 5 '12 at 18:08
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@Wuschelbeutel Kartoffelhuhn The probability is that two consecutive $a$s (the numerator) appear given that either two consecutive $a$s or one $b$ appears (the denominator, including the probability of one $a$ followed by one $b$, which is a win for $b$). You could instead calculate a rather more complicated infinite series but you would get the same answer. –  Henry Oct 5 '12 at 20:09
    
Thanks for the follow-up. I guess the denominator represents the "stopping conditions" after which the whole process is renewed. I was just confused at first that one can use the first two trials as a representation for the entire process, but it makes more sense now. –  Wuschelbeutel Kartoffelhuhn Oct 6 '12 at 8:20

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