I'm looking for help with:
- Confirming or correcting my calculation of the conditional probability when the conditional event is uncertain, and
- How to best utilize probability trees to visualize and understand conditional probabilities when the conditional event is uncertain
Using the following example tree:
Bayes Theorem is just conditional probability when the conditional event is certain. Using the example tree, given that conditional event B has occurred with 100% certainty, we can use Bayes Theorem to get $$P(A|B) = \frac{P(B|A)\cdot P(A)}{P(B)} = \frac{P(B|A)\cdot P(A)}{P(B∩A)+P(B∩¬A)} = \frac{0.1\cdot0.8}{0.08+0.18} = 0.31$$ or we can visualize the P(¬B∩A) and P(¬B∩¬A) branches being eliminated from the probability tree, and use normalization to arrive at what amounts to the exact same calculation $$P(A|B) = \frac{P(B∩A)}{P(B∩A)+P(B∩¬A)} = \frac{0.08}{0.08+0.18} = 0.31$$
However, now I want to use this same tree to determine P(A) and P(¬A) given that B has occurred with less than 100% certainty (call this B', so that we're calculating P(A|B') and P(¬A|B')).
As an example of how this could arise, consider the classic scenario of drawing balls from jars. A is the "red jar" event, so that the probability tree is telling us that if we randomly select a jar to draw a ball from, there is an 80% chance of drawing from a red jar and a 20% chance of drawing from a non-red jar. B is the "black ball" event, so that the probability tree is telling us that if we draw from a red jar, there is a 10% chance that the ball will be black (and thus a 90% chance that it will be something other than black) and if we draw from a non-red jar, there is a 90% chance that the ball will be black (and thus a 10% chance that it will be something other than black).
If we randomly pull a ball from a random jar, and we are using a ball color identifying machine that is only 75% accurate, what is the calculation to determine the probability that we pulled from a red jar given the result from the ball color identifying machine?
I think that the solution is to normalize the cumulative probabilities so that P(B') = 0.75 and P(¬B') = 0.25. Then to get the conditional probability P(A|B'), you would sum the normalized cumulative probabilities under the A branch, and likewise for P(¬A|B'). This makes sense and is a generalized form that is equivalent to Bayes Theorem when P(B') = 1.0 and P(¬B') = 0. That said, I am not 100% confident in this solution (no pun intended) as I could not locate any corresponding examples or explanations.
Now I also want to be able to apply this to the probability tree (assuming the technique is correct) to aid in visualizing and understanding the situation. One thought I had for the presentation was to define the normalization constants C(P(B')) and C(P(¬B')), and append them to the branches along with the resulting normalized cumulative probability. For P(B') = 0.75, this gives $$C(P(B')) = C(0.75) = \frac{P(B')}{P(B)} = \frac{P(B')}{P(B∩A)+P(B∩¬A)} = \frac{0.75}{0.08+0.18} = 2.88$$ $$C(P(¬B')) = C(0.25) = \frac{P(¬B')}{P(¬B)} = \frac{P(¬B')}{P(¬B∩A)+P(¬B∩¬A)} = \frac{0.25}{0.72+0.02} = 0.34$$ $$P(B'∩A) = C(P(B'))\cdot P(B∩A) = 2.88\cdot 0.08 = 0.23$$ $$P(¬B'∩A) = C(P(¬B'))\cdot P(¬B∩A) = 0.34\cdot 0.72 = 0.24$$ $$P(B'∩¬A) = C(P(B'))\cdot P(B∩¬A) = 2.88\cdot 0.18 = 0.52$$ $$P(¬B'∩¬A) = C(P(¬B'))\cdot P(¬B∩¬A) = 0.34\cdot 0.02 = 0.01$$ and the corresponding tree is as follows:
Now just sum the branch's cumulative probabilities to get the conditional probabilities: $$P(A|B') = P(B'∩A)+P(¬B'∩A) = 0.23 + 0.24 = 0.47$$
$$P(¬A|B') = P(B'∩¬A)+P(¬B'∩¬A) = 0.52 + 0.01 = 0.53$$
By giving our B event uncertainty at 75% accuracy, we've moved the conditional probability of A from 31% (using Bayes and 100% accuracy) to 47%, which makes sense because the prior probability of A was higher.
I don't want to reinvent the wheel but I have not been able to locate any guidance on this kind of scenario. So, can you please confirm that the calculation is correct? And does the updated tree make sense as a way to visualize and present the situation? Or do you know of any resources that address this scenario in a more standardized way? Thanks in advance.
Best Answer
Your book most likely wants you to take the values you calculated ($\mathsf P(A'\mid B')=\mathsf P(A\mid B)$, et cetera), and use $\mathsf P(B')=0.75$ to calculate $\mathsf P(A')$.
So use the Law of Total Probability.
$$\mathsf P(A')=\mathsf P(A'\mid B')\mathsf P(B')+\mathsf P(A'\mid \neg B')\mathsf P(\neg B')$$