4.3.2 Conditional Probability

Conditional Probability

Conditional probability is where the probability of an event happening can vary depending on the outcome of a prior event
The event A happening given that event B has happened is denoted A|B
A common example of conditional probability involves selecting multiple objects from a bag without replacement

The probability of selecting a certain item changes depending on what was selected before

This is because the total number of items will change as they are not replaced once they have been selected

Some conditional probabilities can be calculated by using counting outcomes
- Probabilities without replacement can be calculated like this
- For example: There are 10 balls in a bag, 6 of them are red, two of them are selected without replacement
  - To find the probability that the second ball selected is red given that the first one is red count how many balls are left:
  - A red one has already been selected so there are 9 balls left and 5 are red so the probability is $\frac{5}{9}$
You can use sample space diagrams to find the probability of A given B:
- reduce your sample space to just include outcomes for event B
- find the proportion that also contains outcomes for event A
There is a formula for conditional probability that you should use
- $P (A | B) = \frac{P (A \cap B)}{P (B)}$
- This is given in the formula booklet
- This can be rearranged to give $P (A \cap B) = P (B) P (A| B)$
- By symmetry you can also write $P (A \cap B) = P (A) P (B| A)$

If $A$ and $B$ are two events then they are independent if:
- $P (A | B) = P (A) = P (A | B')$
Equally you can still use $P (A \cap B) = P (A) P (B)$ to test for independence
- This is given in the formula booklet

Let $R$ be the event that it is raining in Weatherville and $T$ be the event that there is a thunderstorm in Weatherville.

It is known that $P (T) = 0.035$ , $P (T \cap R) = 0.03$ and $P (T | R) = 0.15$ .

Find the probability that it is raining in Weatherville.

4-3-2-ib-aa-sl-conditional-prob-a-we-solution

State whether the events

R

and

T

are independent. Give a reason for your answer.

4-3-2-ib-aa-sl-conditional-prob-b-we-solution