PS - Independent Events Lesson

Independent Events

Two events are independent of each other if the outcome of one does not affect the probability of the other. Below are two classic examples. One is independent, the other is dependent.

Independent	Dependent

You roll a die and flip a coin.	Draw a ball out of a bag and without replacing it, draw out another ball.
Your probability of flipping "tails" is the same (1/2) no matter what you roll on the die.	This is dependent because your probabilities on your second draw are affected by your first draw. (If you replace the first ball, it becomes independent.)

You can also tell if two events are independent by using the multiplication rule we discussed earlier.

Multiplication Rule for independent events
P(A and B)=P(A)*P(B)

This rule only works if the two are independent. So, if it does not hold true, they are not independent!

Sometimes you don't know whether two events are independent or not. Maybe that is why you are looking at data in the first place. Consider the question "Is preference in ice-cream flavor independent of gender?" View the following presentation to learn more about independence and two way tables.

Chocolate
Female: 4732
Male: 3588
Total: 8320

Vanilla
Female: 4024
Male: 2422
Total: 6466

Total Female: 8756
Total Male: 6010
Total Participants: 14766

This is the list of terms I will use for the proof below.

FC = Female like chocolate

FV= Female like Vanilla

TC = Total like Chocolate

TV = Total like Vanilla

MC = Male like Chocolate

MV = Male like vanilla

I want to prove that, if what is being shown in the video is true when the two events are independent, then this ratio within the joint frequency is also true. Therefore, all the corresponding ratios in the joint frequency should also be equal.

I need to prove, (male like Chocolate)/(female like chocolate)=(male like vanilla)/(Female like Vanilla) or MC/FC=MV/FV

We know this is true from the video

FC/TC=FV/TV

FV=(TV× FC)/TC---(1)

And we also know this ratio is true

MC/TC=MV/TV

MV=(TV × MC)/TC---(2)

So, substitute (1) into (2)

MV/FV= (TV ×MC)/TC ×TC/(TV ×FC)

(TV× MC)/TC × TC/(TV × FC)= MC/FC

So we proved that

MC/FC=MV/FV

(male like Chocolate)/(female like chocolate)=(male like vanilla)/(Female like Vanilla)

Meaning, if the relationship of the two events is independent, then the ratio of the joint frequencies should also be close to equal. Remember, sometimes we might not get exact number for the ratio. However, as long the ratios are near equal, we can conclude that two events are independent.

IMAGES CREATED BY GAVS