Solve Conditional Probability Problems Using Two-way Tables

How could we use a two-way table to find a conditional probability?

 

 

The following two-way table tells us about the favourite subject of 180 students.

 

	Maths	English	Science	Total
Boys	62	26	22	110
Girls	38	16	16	70
Total	100	42	38	180

 

What is the probability that a student selected at random prefers English given that she is a girl?

 

Here, we are selecting out of the students who are girls (our condition).

 

From the second row of the table, we can see that there are 70 girls in total.

So there are 70 possible outcomes for our conditional probability (the denominator).

 

We want the probability that a girl selected at random prefers English.

We can see that there are 16 girls who prefer English.

So 16 is the number of desired outcomes (the numerator).

 

Putting it all together, we then get that the probability that a student selected at random prefers English given she is a girl is:

16/70 = 8/35

 

We can write this as P(E $|\; G)\; =\; 8/35$

$We\; would\; read$P(E $|\; G)\; as\; \text{'}probability$that a student selected at random prefers English given that they are a girl as $|\; means\; \text{'}given\text{'}.$

 

In this example, we were given the two-way table, but often you might need to fill it out first - just like we've done in the past!

 

 

Ready to have a go at some questions?!