# Multiplication rule from frequency table, confusion

I am studying the probability multiplication rule but there is something i don't get: I have this frequency table:

frequency table

and say one sample is randomly picked up, if I want to calculate the probability of sampling one individual which is Male AND Infected my understanding is that as the two events are independent it should be: Pmale=43/81 Pinfected=59/81 Pmale AND Infected=Pmale*Pinfected=0.386

but from the frequency table I see the actual number of infected man is 36, and 36/81 gives 0.4444...shouldn't these percentages be equal? what am i missing?

they turn out to be equal if the number of infected and not infected man and woman is the same (i.e inf/not inf man=30 and inf/not inf woman=2). sorry for the probably very basic question and thanks for your time!

• The events aren't independent. The second method you used is the correct one! – Pspl Aug 28 '19 at 15:57

The second method is correct.

The first method is wrong because when you select the man, the infection rate immediately changes. So the infection probability is $$36/43$$ and if you multiply this by the probability of selecting a man, you would get $$0.44444$$, the correct answer (your second method).

Think of it this way:

You have two bags of marbles. Bag A has Red and Green marbles and Bag B contains Blue and Yellow. If you want to calculate of picking a red marble, you calculate the probability of choosing from bag A instead of B and then multiply by the probability of getting Red from Bag A. Contents of bag B would not matter.

One way of checking if the events $$A$$ and $$B$$ are independent is calculate the conditional probability of one of them, let's say $$P(A)$$, and see if is the same as $$P(A|B)$$, or probability of $$A$$ occurs when $$B$$ already happened (actually you should check the in-dependency of two events by the very way you're using to calculate $$P(A\cap B)$$).

By this you easily conclude that, if $$A$$: the individual is male; $$B$$: the individual is infected:

$$P(A)=\frac{43}{81}\neq \frac{36}{59}=P(A|B)$$

So the events aren't independent.