# Confusion in joint occurrence probability?

Let's say, we have a table

$$\begin{array}{c|c|c|} & \text{Smoking} & \text{Non-Smoking} \\ \hline \text{Male} & 70 & 30 \\ \hline \text{Female} & 40 & 10 \\ \hline \end{array}$$

shows the number of patients.

I would like calculate the joint occurrence probability of being a male and being smoking. Moreover, I assume gender and smoking events are independent of each other.

Is the requested probability = $\frac{70+30}{70+30+40+10}.\frac{70+40}{70+30+40+10}$=$\frac{100}{150}.\frac{110}{150}$ ?

OR is it $\frac{70}{150}$ ? 70 is the number in the cell of intersection of M and S.

• If you pick out one of the patients then the probability that it is a smoking man is simply $\frac7{15}$. Also your table is a strong indication that gender and smoking are not independent at all. – drhab Nov 24 '16 at 11:39

We have $P(male) = \frac{100}{150}$ and $P(smoker)=\frac{110}{150}$. If the variables were really independent then $P(male \cap smoker) = P(smoker) \cdot P(male) = \frac{110}{150} \cdot \frac{100}{150}$. However, I think the variables are not independent. In the case of independence, you would expect the value $\frac{100 \cdot 110}{150}$ in the upper left corner of the matrix, i.e. about 73 insted of 70. If they were independent, both of your suggested methods would give the same result.