1.2.1 Conditional Probability

Question 1

What is conditional probability?

Answer 1

The probability of an event occurring given that another event has occurred. It treats the conditioning event as the new sample space.

Question 2

What is the formula for conditional probability P(A | B)?

Answer 2

P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.

Question 3

What is the formula for conditional probability P(B | A)?

Answer 3

P(B | A) = P(A ∩ B) / P(A), provided P(A) > 0.

Question 4

What is the Multiplicative Law of Probability?

Answer 4

P(A ∩ B) = P(A)P(B | A) = P(B)P(A | B) since P(A ∩ B) = P(B ∩ A)

Question 5

How does conditioning affect the sample space?

Answer 5

It reduces the sample space to the conditioning event, focusing only on outcomes where that event occurs.

Question 6

How are conditional probabilities interpreted in Venn diagrams?

Answer 6

They are the ratio of the overlapping area (A ∩ B) to the conditioning event’s area (B or A).

Question 7

What is an intuitive way to think about P(A | B)?

Answer 7

It’s the fraction of times A occurs among the times B occurs.

Question 8

Why are tree diagrams useful in conditional probability?

Answer 8

They organize sequential probabilities and clearly show how each condition affects the outcome.

Question 9

What do Venn diagrams help illustrate in conditional probability?

Answer 9

How the conditional event redefines the sample space, highlighting overlap.

Question 10

When is it useful to use a table to analyze conditional probability?

Answer 10

When there are two categorical variables with multiple outcomes, tables organize joint and marginal probabilities well.