What is Conditional Probability?

Here is a die, and a bag of disks.

Inside the bag are 8 disks – 3 red, 5 blue.

Now, let’s look at the probabilities of 2 different events.

First, the probability of pulling a blue disk from the bag is…

The probability of pulling a blue disk from the bag is 5 over 8.

Second, the probability of rolling a 5 on the die is…

The probability of rolling a 5 on the die is 1 over 6.

Now, suppose we pull out a red disk…

Leaving 2 red disks and 5 blue disks.

We can now ask about the probabilities of the same 2 events again…

But this time, we’re equipped with the knowledge that a red disk has been taken from the bag.

So now… given that we’ve pulled a red disk from the bag…

The probability of pulling a blue disk is…

The probability of pulling a blue disk is now 5 over 7.

Next… given that we’ve pulled a red disk from the bag…

The probability of rolling a 5 is…

The probability of rolling a 5 is still just 1 over 6.

Now, statisticians call this second set of probabilities…conditional probabilities.

And that’s because, for each of these, we’re asking ‘what is the probability of one event…given another has occurred’. 

Or, put another way, ‘what is the probability of one event… on the condition that another event has occurred’.

So now, a conditional probability is…

A conditional probability is the probability of one event…

On the condition that another event has occurred.

Which means that… if we want the probability of landing on Heads, on the condition that a die roll landed on a 4… 

Then the probability of landing on Heads is a conditional probability.

And… if we want the probability of picking a yellow marble from a bag, on the basis that we previously picked a blue marble…

Then the probability of picking a yellow marble is a conditional probability.  

Now, in each example so far, the conditional probability has been connected to a second action:

We’ve picked a disk from a bag, and then… taken another disk…or even rolled a die.

But, actually, we can have conditional probabilities for single actions, too. 

For example, we could ask: 

‘If I roll a die once, what is the probability that I’ve rolled a 4… on the condition that I’ve rolled an even number’.

And that is still a conditional probability!

So now, which of the following describes a conditional probability?

These all describe conditional probabilities. 

So, to sum up: 

When we want to know the probability of one event, on the condition that another event has occurred, we call this…

When we want to know the probability of one event, on the condition that another event has occurred, we call this… conditional probability.

And we can find conditional probabilities for…

We can find conditional probabilities for single actions… or… multiple actions.