Reading a conditional probability off a two-way table

“Given that” does one thing: it shrinks the denominator. You stop dividing by everybody and start dividing by the group you were handed.

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 Recognize the Question

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

Given a row

160 commuters were surveyed by mode and punctuality.

Bus: 54 on time, 46 late. Bicycle: 45 on time, 15 late.

What is the probability a commuter was on time, given that they cycled? Round to the nearest hundredth.

P =

Given a column

Same 160 commuters. Bus: 54 on time, 46 late. Bicycle: 45 on time, 15 late.

What is the probability a commuter took the bus, given that they were late? Round to the nearest hundredth.

P =

Both directions, side by side

Same 160 commuters. Bus: 54 on time, 46 late. Bicycle: 45 on time, 15 late.

What is the probability a commuter was late, given that they took the bus? Round to the nearest hundredth.

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

\(P(A \mid B) = \dfrac{\text{cell}}{\text{total of } B}\) the given group becomes the whole world

\(P(A \mid B) \neq P(B \mid A)\) same cell on top, different total underneath

Find the cell as usual. Then ask what you were told to assume, and use that row or column total as the denominator. The numerator is the same cell either way — only the bottom of the fraction moves, and it moves to whichever group the word “given” points at.

Common slips

(1) Divided by the grand total. That is the plain “and” probability, not a conditional. (2) Conditioned on the wrong one. P(improved given standard) and P(standard given improved) use different denominators and are different numbers. (3) Assumed the two must be close. They can be far apart, and being close is not evidence you did it right.

04 Practice — Step by Step

Shrink the denominator

A clinic recorded the outcome for 250 patients, half on the standard treatment and half on a new one.

One patient record is chosen at random. Round to the nearest hundredth.

Step 1

Given the row

What is the probability a patient improved, given that they were on the standard treatment?

P =

Step 2

The same cell, the other way round

Now: what is the probability a patient was on the standard treatment, given that they improved?

P =

Step 3

Which denominator does this one want?

“What is the probability a patient was on the standard treatment and improved?” Which total goes underneath?

Q3 Q5

Reading a conditional probability off a two-way table

Three answers, one cell