Using two conditional probabilities to answer a yes-or-no question

The previous question asked you to compute one conditional. This one asks you to compute two and then say what they mean — and the marks are in the second half.

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.

A clear difference

220 seedlings were grown with or without a fertiliser.

Fertilised: 84 flowered, 36 did not. Unfertilised: 35 flowered, 65 did not.

What is the probability a seedling flowered, given that it was fertilised? Round to the nearest hundredth.

P =

The groups are different sizes

Same 220 seedlings. Fertilised: 84 flowered, 36 did not (120 in total). Unfertilised: 35 flowered, 65 did not (100 in total).

A classmate says “84 flowered with fertiliser and only 35 without, so fertiliser more than doubles flowering.” Is that reasoning sound? Enter 1 for yes, 0 for no.

Barely any difference

250 customers were shown one of two checkout layouts.

Layout A: 39 completed, 61 abandoned. Layout B: 59 completed, 91 abandoned.

What is the completion rate for Layout B? 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

compare \(P(A \mid B)\) with \(P(A \mid \text{not } B)\) same outcome, the two different groups

far apart → associated; close → not the raw counts cannot tell you this

Pick the outcome you care about. Work out its rate inside one group, then inside the other group. Because both are rates, the groups can be different sizes and the comparison still holds — which is exactly why you cannot answer this by comparing the raw counts.

Common slips

(1) Compared counts instead of rates. 90 promoted against 57 means nothing until you know the group sizes. (2) Reversed one of them. Both conditionals must condition on the group and ask about the outcome, not the other way round. (3) Answered the arithmetic and stopped. The question ends with a yes or a no, and leaving it off leaves marks on the table.

04 Practice — Step by Step

Two rates, then a verdict

A company looked at whether its optional training course was associated with promotion. It recorded 300 employees.

Round each probability to the nearest hundredth.

Step 1

The rate inside the first group

What is the probability an employee was promoted, given that they attended the training?

P =

Step 2

The same rate inside the other group

And the probability an employee was promoted, given that they did not attend?

P =

Step 3

Now say what it means

Does the data suggest attending the training is associated with being promoted?

Q16 Q18

Using two conditional probabilities to answer a yes-or-no question

What this does and does not show