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I was reading this example about finding the probability of one actually getting the disease, after he is diagnosed positive by a test, as he is determining whether or not to proceed with treatment as the treatment has great side effects.

Let's say P(A) = Probability of getting the disease and P(B) = Probability of getting tested positive

In the example, in order to find the probability of one actually getting the disease when he is tested positive is a conditional probability of P(A|B). However, in my opinion, this situation can also be represented by P(A and B). To me, P(A and B) means the probability of getting the disease AND ALSO getting tested positive. Isn't this also an accurate representation on what the question wants?

P(A|B) means the chance of him really infected by the disease given that he has been tested positive. But P(A and B) means that he is both infected by the disease as well as being tested positive. However, the calculated values of P(A|B) and P(A and B) are quite different..

I'd really appreciate any input here as I'm trying to get my basic concepts in probability right.

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I think probability are differents because once you now B is true(tested positive) the population you are going to measure A is B, then probability change.

In other case, you measure probabilty of A AND B over all the population

For example if you flip a coin twice possible results are (CX, CC, XC, XX)

P(C and X) = 1/2 (cx,xc) out of (CX, CC, XC, XX)

On the other hand

P(C|X) = 2/3 (cx,xc) out of (CX, XC, XX)

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Reason that they are different is that in $P(A|B)$ you KNOW that B happened, and you're only measuring probability of A with renormalized sample space B.This means that your sample space is not $\Omega$ anymore, but B since you know B happened, and you're restricing yourself to B only. So, your new sample space is B and you're measuring probability of A over B(Not A over $\Omega$ as in $P(A)$).In $P(AB)$ however, you don't know anything, and you're measuring probability that both A and B will happen.

Reason why you use $P(A|B)$ is in part of the sentence "...after he is diagnosed positive on the test, ...". So you KNOW that he already is diagnosed positive and you want to see probability that he actualy have disease GIVEN his positive results. $P(AB)$ would be applicable in case that you need to compute probability of having disease AND getting positive results(you don't know nothing about him, wheather he acualy have disease or not, or if he is tested positive or not).

Hardest part about probability is intuition :D

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These are related but very different probabilities.

Think of a jar with three balls in it: a red ball, a blue ball, and a green ball. Suppose you draw two balls without replacement at random and we want to know the probability that you pull a red and a blue ball.

$P(\text{R and B}) = P(\text{R or B on first roll}) * P(\text{R or B on second roll}) = 2/3 * 1/2 = 1/3$

Now think of the same situation - one red, one blue, one green ball in a jar. Suppose you've already pulled out a red. Now, what is the probability that you pull out a blue?

$P(\text{B|R}) = P(\text{B given that you have already pulled R}) = 1/2$

In both of these cases, we're evaluating a probability regarding the event that a blue ball is pulled and the event that a red ball is pulled. That's perhaps where your confusion arises; it seems very much like these two things are the same. However, we are evaluating these probabilities at different times. The first probability above (the joint probability) is evaluated before we pull anything and before we have any additional information. The second probability above (the conditional probability) is evaluated once we have pulled the first ball and thus is evaluated when we have more information.

Joint probabilities, roughly speaking, are the probabilities of two (or more) events occurring when you don't have any additional information. Conditional probabilities are the probabilities concerning an event evaluated when you do have additional information relative to a joint or marginal probability.

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