Conditional Probabilities for a Race? - Sequence of Dependent Events?

Question

I've been getting stumped by what seems to be a super simple conditional probability question...

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3 people run a race: A, B, and C. Based on what we know about about their past races, we estimate the probability of each person coming in $1^{st}$, $2^{nd}$, and $3^{rd}$ are...

|   |  1st  |  2nd  |  3rd  |
|---|-------|-------|-------|
| A |  0.5  |  0.3  |  0.2  |
| B |  0.3  |  0.2  |  0.5  |
| C |  0.2  |  0.5  |  0.3  |

1. How likely is it that A runs $1^{st}$ and B runs $2^{nd}$
2. Given that Person A wins the race, what is the probability that person B comes in second?

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Here's the main part that's confusing me...

Shouldn't $P(B=2|A=1) = P(C=3|A=1)$ since the race will be deterministic as soon as we know 2 of the 3 finishers? I just can't seem to find a way to calculate these probabilities that satisfies that criterium.

(Note: It's also possible that this is a trick question with impossible probabilities, in which case I'd really appreciate a derivation that can support this.)

Answer 1

There is more than one possible solution. Here are two different scenarios, and any weighted combination of them works too

Order   Scenario 1  Scenario 2
ABC         0.0         0.2
ACB         0.5         0.3
BCA         0.0         0.2
BAC         0.3         0.1
CAB         0.0         0.2
CBA         0.2         0.0

The answer to (1) is the probability $\mathbb P(ABC)$, so between $0$ and $0.2$

The answer to (2) is the conditional probability $\frac{\mathbb P(ABC)}{\mathbb P(ABC)+\mathbb P(ACB)}$, i.e. the answer to (1) divided by the probability $A$ wins $(0.5)$, so between $0$ and $0.4$