# Conditional Probability and the Probability of "AND" & "OR"

- Author:
- dthurston, Micky Bullock

In the following diagram:
The circle on the left represents the part of the sample space in event A
The circle on the right represents the part of the sample space in event B.
This means the overlap of the two circles represents the part of the sample space in BOTH event A and event B (a.k.a AnB!)
This means that the total space covered by the two circles represents the part of the sample space in AT LEAST ONE OF event A and event B (a.k.a. AuB!)

**Day 1:**Click the button for "Day 1". Let's examine the overlap. We can find out how much is in this space by either -

- (1) Looking at how much is in the green space; (2) then looking at how much of that is also covered by red. OR
- (1) Looking at how much is in the red space; (2) then looking at how much of that is also covered by green.

**Day 2:**Unclick the button for Day 1 and click the button for Day 2 (only click the hint later if you need it!) Let's look at the total space covered... First -> Use the longest slider to pull the two circles completely apart! Q1: How can we use the probabilities on the screen to find out how much TOTAL is covered by these circles? Q2: Move the circles using the slider so that the method you described for #1 would no longer work. Why won't this work? Q3: How could we adjust the method in #1 to find the total amount covered by the two circles now? Q4: What do we call the space covered by AT LEAST ONE of these events?

*Conclusions:*How can we use the probabilities above to find P(A and B)? How can we use the probabilities above to find P(A or B)?