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part of a question that I am trying to asks to "use the definition of conditional probability along with results of parts (b) and (c) to calculate P(has disease/positive test)".

Now I am confused as to why I would need to use conditional probability and weather or not P(A/B) is division of P(A)/P(B).

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    One to watch out for: some people (bad, bad people) might write $P(A\backslash B)$ to mean the probability of $A$ and not $B$, interpreting the $\backslash $ to mean set difference.2012-03-01

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$P(A|B)$ is the probability of $A$, given that $B$ has already occurred.

This is not the same as $\frac{P(A)}{P(B}$.

In fact $\displaystyle P(A|B) = \frac{P(A \cap B)}{P(B)}$

Given the phrasing about diseases and tests, I am quite confident by $P(A/B)$ they actually mean $P(A|B)$.

As to why conditional probability needs to be used here is an example.

Suppose there are 50 girls and 50 boys in a class. Out of the 50 boys, 30 have black hair and 20 are blonde. Out of the 50 girls, 40 are blonde and 10 have black hair.

Now I pick a random child, and see that the hair is black. What is the probability that it is a girl?

If you take the two events:

  • A: The child picked is a girl.

  • B: The child picked has black hair.

The probability that is being asked is the conditional probability $P(A | B)$.

If you try to reason it out, without any probability formulas, you can reason as:

There are 40 children with black hair. Out of them, 10 are girls.

Thus if the child I picked has black hair, the chance that it is a girl is $\frac{10}{40}$.

Basically, because the child you picked has black hair, your space of possibilities has reduced and you compute the probabilities based on that new information.

This is exactly what the formula $\displaystyle P(A|B) = \frac{P(A \cap B)}{P(B)}$ describes mathematically.

Here $P(B) = \frac{40}{100}$ and $P(A \cap B) = \frac{10}{100}$, and $P(A|B) = \frac{10}{40}$.

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It is more customary to write $P(A \mid B)$ rather than $P(A/B)$ for conditional probability, but I'm quite certain the author intended conditional probability. This problem is likely asking you to write $ P(A \mid B) = \frac{P(A \cap B)}{P(B)} $ (the definition of conditional probability), then plug in some values that you found in earlier parts of the question.