Cogito
Integrated Math 2 · Chapter 8 · Lesson 2
Conditional Probability
Given that something already happened.
12 problems · about 21 minutes · S-CP.A.3, S-CP.B.6
What this lesson teaches
The student computes conditional probabilities and tests events for independence.
- P(A given B) = P(A and B) ÷ P(B).
- Conditioning replaces the sample space with a slice of it.
- If conditioning changes nothing, the events are independent.
Warm Up
Straightforward practice. Get the method working first.
5 problemsP(A and B) = 0.18, P(B) = 0.6. What is P(A given B)?
Answer 0.3
Why 0.3.
What does conditioning on B do?
Answer It shrinks the sample space to just B.
Why B becomes the new whole.
P(A and B) = 0.3, P(B) = 0.6. P(A given B)?
Answer 0.5
Why 0.3 ÷ 0.6.
P(A and B) = 0.12, P(B) = 0.4. P(A given B)?
Answer 0.3
Why 0.12 ÷ 0.4.
18 of 24 in a group have a trait. P(trait given the group)?
Answer 0.75
Why 18 ÷ 24.
Build It Up
The same ideas with more to keep track of.
3 problemsP(A given B) = P(A). Independent? 1 yes, 0 no.
Answer 1
Why The condition changed nothing.
A die shows an even number. P(it is a 6), to three decimal places?
Answer 0.333
Why 1 of 3 even faces.
Is P(A given B) always equal to P(B given A)? 1 yes, 0 no.
Answer 0
Why The denominators differ.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the conditional probability method in order.
Answer 1. Identify which event is the condition. 2. Use that event as the new total. 3. Count how many of those also satisfy the other event. 4. Divide to get the conditional probability.
Why The condition sets the denominator.
P(A and B) = 0.35, P(B) = 0.7. P(A given B)?
Answer 0.5
Why 0.35 ÷ 0.7.
The Condition: P(A and B) = 0.2 and P(B) = 0.5. What is P(A given B)?
Answer 0.4
Why 0.4.
The Table: Of 60 sport players, 45 play music. What is P(music given sport)?
Answer 0.75
Why 0.75.