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"Either/Or" Probability with Venn Diagrams and Two-Way Tables

Author: Sophia
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what's covered
In this tutorial, you're going to learn about solving “either/or” probability for overlapping events in Venn diagrams and tables. Our discussion breaks down as follows:

Table of Contents

1. “Either/Or” Probability for Overlapping Events With Venn Diagrams

Let's see how we can calculate the “either/or” probability for overlapping events using data from Venn Diagrams.

hint
Venn diagrams are particularly helpful for visualizing why you need to subtract the overlapping probability for these calculations.

EXAMPLE

A Venn diagram with two overlapping circles, one labelled 'A' and the other labelled 'B'. The overlapping region contains the number 0.12, the region inside circle A but outside circle B contains the number 0.30, the region outside circle A and inside circle B contains the number 0.27, and the region outside the circles contains the number 0.31.
Using this Venn diagram, find the probability that event A or event B occurs.
Here we have overlapping events, so we can note that:
P left parenthesis A space o r space B right parenthesis equals P left parenthesis A right parenthesis plus P left parenthesis B right parenthesis minus P left parenthesis A space a n d space B right parenthesis
The probability that A happens is the whole circle of A, or:
P left parenthesis A right parenthesis equals 0.30 plus 0.12 equals 0.42
The probability that B happens is the whole circle of B, or:
P left parenthesis B right parenthesis equals 0.27 plus 0.12 equals 0.39
The probability that A and B happens is the overlap of the Venn diagram, or:
P left parenthesis A space a n d space B right parenthesis equals 0.12
Plugging these values into the formula, we get:
P left parenthesis A space o r space B right parenthesis equals 0.42 plus 0.39 minus 0.12 equals 0.69
You can also note that if you simply add up all the parts, 0.30 + 0.12 + 0.27, you can also get the final result.

try it
A Venn diagram with two overlapping circles, one labelled 'A' and the other labelled 'B'. The overlapping region contains the number 0.11, the region inside circle A but outside circle B contains the number 0.43, the region outside circle A and inside circle B contains the number 0.23, and the region outside the circles contains the number 0.23.
Using this Venn diagram, what is the probability that event A or event B occurs?
To find the probability that event A or event B occurs, we can use the following formula for overlapping events:

P left parenthesis A space o r space B right parenthesis equals P left parenthesis A right parenthesis plus P left parenthesis B right parenthesis minus P left parenthesis A space a n d space B right parenthesis equals 0.54 plus space 0.34 minus 0.11 equals 0.77

The probability of event A is ALL of circle A, or 0.43 + 0.11 = 0.54.
The probability of event B is ALL of circle B, or 0.23 + 0.11 = 0.34.
The probability of event A and B is the intersection of the Venn diagram, or 0.11.

We can also simply add up all the parts, 0.43 + 0.11 + 0.23 = 0.77.

2. “Either/Or” Probability for Overlapping Events With Two-Way Tables

Let's look at another example using a two-way table, shown below. Students in the middle school were asked about their dominant hand. Some are right-handed sixth graders, left-handed sixth graders, etc.


Dominant Hand
Right Left Ambidextrous
Grade 6th 99 9 2 110
7th 90 31 0 121
8th 93 11 3 107


282 51 5 338

What's the probability that a student is either in eighth grade or left-handed?

The probability of being in eighth grade or left-handed is equal to the probability of being an eighth grader plus the probability of being a left-handed student minus the probability of both.

table attributes columnalign left end attributes row cell P left parenthesis 8 space o r space L right parenthesis equals P left parenthesis 8 right parenthesis plus P left parenthesis L right parenthesis minus P left parenthesis 8 space a n d space L right parenthesis end cell row cell P left parenthesis 8 space o r space L right parenthesis equals 107 over 338 plus 51 over 338 minus 11 over 338 end cell row cell P left parenthesis 8 space o r space L right parenthesis equals 147 over 338 end cell end table
Why minus the probability of both? Because you counted the left-handed eighth graders in the eighth-grade row and in the left-hand column. We double counted those 11 students, and you only want to count them once. Add and subtract those probabilities as shown below, and you end up with 147/338.

The other way to approach this is by simply adding up the cells that are either left-handed or eighth grade.


Dominant Hand
Right Left Ambidextrous
Grade 6th 99 9 2 110
7th 90 31 0 121
8th 93 11 3 107


282 51 5 338

You would add up all the left-handed sixth graders, left-handed seventh graders, left-handed eighth graders, right-handed eighth graders, and ambidextrous eighth graders. These added together equal 147 out of the total number of students, 338.

try it

 Dietary Preference
Meat-Eater Vegetarian Vegan
Hair Texture Curly Hair 80 30 10 120
Straight Hair 100 50 20 170

180 80 30
According to these data, what is the probability a person has curly hair or eats meat?
P left parenthesis C u r l y space o r space M e a t right parenthesis space equals space P left parenthesis C u r l y right parenthesis space plus space P left parenthesis M e a t right parenthesis space minus space P left parenthesis C u r l y space a n d space M e a t right parenthesis
P left parenthesis M e a t space o r space C u r l y right parenthesis space equals space 180 over 290 space plus space 120 over 290 space minus space 80 over 290 space equals space 220 over 290 space o r space 0.76 space o r space 76 percent sign

If a student is selected at random, what is the probability that the student is a girl who chose apple as her favorite fruit? Answer choices are rounded to the hundredth place.

summary
“Either/or” probability for overlapping events is calculated by adding the probabilities of the two events, then subtracting their joint probability. The reason you subtract out the joint probability is that it is counted in both of their individual probabilities. You don't want to count it twice; you only want it counted once. You explored examples of visual representations of “either/or” probabilities for overlapping events utilizing both Venn diagrams and two-way tables.

Good luck!

Source: THIS TUTORIAL WAS AUTHORED BY JONATHAN OSTERS FOR SOPHIA LEARNING. PLEASE SEE OUR TERMS OF USE.

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