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The following table represents all possible paths of dichotomous events at 5 time moments. At each time moment either 1 or -1 event occurs with probabilities $p$ and $q$. Time stops when one observes 3 of either 1 or -1 events. So time can stop after 3, 4 or 5 moments. I am interested in a probability distribution of the number of time moments - $N$.

    T1 T2 T3 T4 T5
 1: -1 -1  1  1  1
 2: -1  1 -1  1  1
 3:  1 -1 -1  1  1
 4: -1  1  1 -1  1
 5:  1 -1  1 -1  1
 6:  1  1 -1 -1  1
 7: -1  1  1  1   
 8:  1 -1  1  1   
 9:  1  1 -1  1   
10:  1  1  1      
11:  1  1 -1 -1 -1
12:  1 -1  1 -1 -1
13: -1  1  1 -1 -1
14:  1 -1 -1  1 -1
15: -1  1 -1  1 -1
16: -1 -1  1  1 -1
17:  1 -1 -1 -1   
18: -1  1 -1 -1   
19: -1 -1  1 -1   
20: -1 -1 -1

For example, before time starts, the probability of $N = 3$ is $p^3+q^3$ and $P(N=4)=3p^3q+3q^3p$.

I struggle with conditional probabilities at different time moments. For example, what is $P(N=4|T_3,C)$ - probability after time moment 3 with condition that the sum of three events is say -1. Or $P(N=5|T_2,C)$ - probability after time moment 2 with condition that the sum of two events is 0.

I am interested in a formula for $P(N=k|T_j,C_j)$ where $C_j$ is the sum of event outcomes up to time moment $j$. And would also like to see a walk-through example.

What is exact framework or formula for computing such probabilities and what are the answers for these examples?

whuber
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danas.zuokas
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  • It's unclear what you mean by "such probabilities:" are you referring to *this particular problem?* If so, your enumeration of the outcomes coupled with your calculation of the probabilities already provides the framework. But if you mean problems that are somehow similar to this one, then please tell us in what ways they are similar. That is, how do you propose generalizing this example? – whuber May 11 '19 at 14:00
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    I am interested in a formula for $P(N=k|T_j,C_j)$ where $C_j$ is the sum of event outcomes up to time moment $j$. And would also like to see a walk-through example. – danas.zuokas May 13 '19 at 05:52
  • Why can't this process continue after $5$ time steps? What conditions are you imposing that causes this? – whuber May 13 '19 at 15:23
  • It could be any system. Think of basketball playoffs, for example. What is the probability of seeing 5 games after 1-1 draw? – danas.zuokas May 14 '19 at 05:32
  • Do you mean your actual question might be something like the one posed at https://stats.stackexchange.com/questions/329521? – whuber May 14 '19 at 13:36
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    Yes it is similar. But I am interested not in a probability to win, but in probability distribution of the possible number of games. – danas.zuokas May 15 '19 at 05:38
  • Good! Could you please edit your post to make it more evident that this is your aim? – whuber May 15 '19 at 15:01

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