0.11 Independent and dependent events

Event $A$ is rolling a 1 and event $B$ is rolling a 6. Since the outcome of the first event does not affect the outcome of the second event, the events are independent.

The probability of rolling a 1 is $\frac{1}{6}$ and the probability of rolling a 6 is $\frac{1}{6}$ .

Therefore, $P\left(A\right)=\frac{1}{6}$ and $P\left(B\right)=\frac{1}{6}$ .

The probability of rolling a 1 and then rolling a 6 on a fair die is $\frac{1}{36}$ .

Event $A$ is selecting a R1 coin and event $B$ is next selecting a R2. Since the outcome of the first event affects the outcome of the second event (because there are less coins to choose from after the first coin has been selected), the events are dependent.

The probability of first selecting a R1 coin is $\frac{1}{4}$ and the probability of next selecting a R2 coin is $\frac{2}{3}$ (because after the R1 coin has been selected, there are only three coins to choose from).

Therefore, $P\left(A\right)=\frac{1}{4}$ and $P\left(B\right)=\frac{2}{3}$ .

The same equation as for independent events are used, but the probabilities are calculated differently.

The probability of first selecting a R1 coin followed by selecting a R2 coin is $\frac{1}{6}$ .

P(male).P(positive result)= $\frac{120}{210}=0,57$

P(female).P(positive result)= $\frac{90}{210}=0,43$

P(male and positive result)= $\frac{50}{210}=0,24$

P(male and positive result) is the observed probability and P(male).P(positive result) is the expected probability. These two are quite different. So there is no evidence that the medication's success is independent of gender.

To get gender independence we need the positive results in the same ratio as the gender. The gender ratio is: 120:90, or 4:3, so the number in the male and positive column would have to be $\frac{4}{7}$ of the total number of patients responding positively which gives 51,4. This leads to the following table: