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Suppose there are two coins A and B. When tossing a coin $i$, "head" happens with probability $p_i$.

The problem is that $p_i$ itself is a random variable. Say that the associated probability density function is given by $f_i$ for coin $i$.

We also know the conditional probability $f_{p_A|p_B=p}$.

In this setting, what is the formula to compute the probability of head for coin A given that coin B showed head? i.e., $P[A:Head|B:Head]$?

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We can use conditional probability formula to get what we want: $$P(A:Head|B:Head)=\frac{P(A:Head\cap B:Head)}{P(B:Head)}$$

First we condition on $p_a,p_b$ (using lowercase for notational simplicity), and use total probability law: $$P(A:Head\cap B:Head)=\int P(A:Head\cap B:Head \ \vert\ p_a,p_b)f_{P_a,P_b}(p_a,p_b)dp_adp_b$$

$P(A:Head\cap B:Head|p_a,p_b)$ simplifies to $p_ap_b$ because we don't need anything else other than the head probabilities, i.e. the events are conditionally independent given the $p_i$'s. Also, the joint density can be written as $f_{P_a,P_b}(p_a,p_b)=f_{P_a|P_b}(p_a|p_b)f_{P_b}(p_b)$. The final expression is something like: $$P(A:Head\cap B:Head)=\int p_ap_b f_{P_a|P_b}(p_a|p_b)f_{P_b}(p_b)dp_adp_b=E[P_aP_b]$$ Similarly, we could find $$P(B:Head)=\int P(B:Head|p_b)f_{P_b}(p_b)dp_b=E[P_b]$$

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  • $\begingroup$ I think you have assumed that the two coins are independent to each other? Is that why you have $P(A:H~and~B:H)=\int p_ap_b f_{p_ap_b}(p_a,p_b)dp_adp_b$? Does this also work for possibly correlated case? $\endgroup$ – Andeanlll Sep 12 '19 at 2:00
  • $\begingroup$ No I haven't assumed it, if they were independent, I'd have written $$P(A:Head|B:Head)=P(A:Head)$$ However, note that "given" their head probabilities, they're independent. $\endgroup$ – gunes Sep 12 '19 at 8:19
  • $\begingroup$ Alright! Thanks for the clarification :) $\endgroup$ – Andeanlll Sep 13 '19 at 4:29
  • $\begingroup$ Sorry for keep posting comments, but one last question for clarification. $P_A$ and $P_B$ are random variables here. So, shouldn't $P_A|P_B$ also be an RV? Can we take $E[P_A|P_B]=\frac{E[P_AP_B]}{E[P_B]}?$ $\endgroup$ – Andeanlll Sep 18 '19 at 3:22
  • $\begingroup$ Yes, $P_A|P_B$ represents a random variable. But, the expectation you wrote is wrong. $\endgroup$ – gunes Sep 18 '19 at 3:56

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