Question

What is the conditional probability of an event and how does it differ from the regular probability of an event occurring?

Answer 1

A conditional probability has a new, but smaller sample space compared to the original probabilty

Explanation:

It is best explained with an example ...

A die has 6 possible outcomes, so if you throw the die the results will be one of the following from the sample space:

6-sided die: `{1,2,3,4,5,6}`

Now, let's condition the probability with the following given piece of information: The result is odd .

Given the toss of the die is odd, the new, but smaller sample space is:

6-sided die, given it is odd: `{1,3,5}`

Under the first scenario, the probability of rolling a 3 is:
`P(3)=1/6`

Under the second conditional scenario, the probability of rolling a 3 is:

`P(3|odd)=1/3`

In summary, under the first scenario, we divided by the sample space of 6 whereas under the conditional probability we had only 3 in the sample space. This will make a big difference in the probability of rolling a 3 as shown above.

Hope this helped you understand the difference!