Computing conditional probability using a sample space

The first conditional probability question on the exam, and the cleanest place to see what the idea actually is: being told something shrinks the pool you are choosing from.

01 Worked Example Video

▸ VIDEO COMING SOON

A short walkthrough explaining what you need to know and how to solve this question type lands here once it's recorded.

02 What This Looks Like on ALEKS

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

Eight cards

Eight cards are numbered 1 to 8. Cards 1, 2, 4, 5, 6, 7 and 8 are grey; card 3 is white.

A card is drawn at random. What is the probability it is card 4, given that it is grey? Write your answer as a fraction.

P =

When the answer is zero

Eight cards are numbered 1 to 8. Cards 1, 2, 4, 5, 6, 7 and 8 are grey; card 3 is white.

What is the probability the card is number 3, given that it is grey? Write your answer as a fraction or a whole number.

P =

Twelve cards

Twelve cards are numbered 1 to 12. Eight of them are grey, including card 5.

What is the probability the card is number 5, given that it is grey? Write your answer as a fraction.

P =

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 The Rule

\(P(A) = \dfrac{\text{ways A happens}}{\text{everything}}\) the ordinary probability you already know

\(P(A \mid B) = \dfrac{\text{ways A happens}}{\text{everything in B}}\) the bar means the denominator changed

The vertical bar reads “given that”. Everything to the right of it is information you have been handed, and its only job is to replace the denominator. You are no longer choosing from everything; you are choosing from the smaller group the condition describes.

Common slips

(1) Kept the original denominator. That is the whole error conditional probability exists to test. (2) Read the bar backwards. P(A | B) conditions on B, not on A. The thing after the bar is what you were told. (3) Forgot the numerator can shrink too. If the item you want is not inside the condition, the answer is 0.

04 Practice — Step by Step

Watch the denominator move

Ten cards are numbered 1 to 10. The cards numbered 1, 2, 3, 5, 8 and 9 are grey; the rest are white.

One card is drawn at random.

Step 1

The plain probability

What is the probability of drawing the card numbered 3? Write it as a fraction.

P =

Step 2

How many are grey?

Count the grey cards.

Step 3

Now condition on grey

What is the probability the card is number 3, given that it is grey? Write it as a fraction.

P =

Q8 Q10

Computing conditional probability using a sample space

That is the whole idea