This is a classic question, and I understand the answer from one approach, but I'm trying to understand it the underlying conditional probabilities.
We have two bags: Bag A contains 5 keys, Bag B contains 7 keys. One of the 12 key's fits a lock. If we select a bag and key at random what is the probability we find the correct key $P(F)$.
Answer: We partition the event space into 4 possibilities, and sum the appropriate events multiplied by the probability of finding the key given that event.
- $A_1$ Key in Bag A, and we Pick A: $P(A_1) = 5/12 • 1/2 = 5/24$
- $A_2$ Key in Bag B, and we Pick A: $P(A_2) = 7/12 • 1/2 = 7/24$
- $B_1$ Key in Bag B, and we Pick B: $P(B_1) = 7/12 • 1/2 = 7/24$
- $B_2$ Key in Bag A, and we Pick B: $P(B_2) = 5/12 • 1/2 = 5/24$
$P(F) = 1/5P(A_1) + 1/7P(B_1) = 1/12$
Questions: I'm struggling to work out the conditional probability's $P(F|A_1)$ & $P(F|A)$: And if i have worked them out correctly, why they make sense:
- $P(F|A_1)$ = $\frac{P(F \cap A_1)}{P(A_1)}$ = $\frac{(1/5)(5/24)}{5/24}$ = $1/5$ = $\frac{P(A_1|F)P(F)}{P(A_1)}$ = $\frac{P(A_1|F)1/12}{5/24}$
- Therefore: $P(A_1|F) = 1/2$
- I think $1/5$ makes intuitive sense, but i'm not happy with my calculations:
- I've basically written: $P(F|A_1) = \frac{P(F|A_1)P(A_1)}{P(A_1)}$ This feels wrong? A tautology?
- $P(A_1|F) = 1/2$ I'm unsure of, and am not finding intuitive.
- I guess $P(A_1|F) = 1/2$ could be saying that if we know we have found the key, then it had to be in either scenario $A_1$ or $B_1$ but given they are weighted differently i'm unclear why $P(A_1|F) = 1/2$
- $P(F|A) = \frac{P(F \cap A)}{P(A)}$ = $\frac{(5/24)(1/5)}{1/2}$ = $1/12$ = $\frac{P(A|F)P(F)}{P(A)}$ = $\frac{P(A|F)1/12}{1/2}$
- Therefore: $P(A|F) = 1/2$
- Again, I think $P(F|A) = 1/12$ makes intuitive sense, as just because we have picked bag A, doesn't make it any more likely, that the Key is in Bag A, and hence doesn't effect the overall probability of finding the key.
- $P(A|F) = 1/2$ like before is bothering me, I suppose the previous logic could apply? We've found the key, which bag did we pick, it has to be one of the two 50/50....But that doesn't feel right?
I'd be very grateful for help deciphering these conditional probabilities and the intuition behind them! Thanks!
Additional Context: Picture of original question and answer.
