In this video we discuss what are complementary events in probability and statistics. We use a Venn Diagram for a visual and cover 3 equations for complementary events that help to calculate probabilities.

Transcript/notes
Complementary events in probability
In probability, the complement of an event is the outcomes in the sample space that are not part of an event. For instance if we were rolling a die, and we assigned the event to be rolling a 1 or a 2, the complement of that event is rolling a 3 through 6.

Our event is written as P of E, probability of event E, and the complement of an event is written as P of E prime with this little apostrophe here.

And there are a few rules or equations that go with complementary events. Looking at the sample space for a die roll, we know that each of these has a probability of 1 over 6, and if we add them up we get 1. So the probability of all outcomes in the sample space equals 1 or 100%.

With that knowledge, the probability of event E plus the probability of E prime equals 1. For instance in our example rolling a 1 or a 2 on a die would be probability of event E, and rolling a 3 through a 6 would be probability of E prime. If we add probability of E and E prime we get 1.

To see this visually, we can look at a Venn diagram. So, here is a square, and the area inside this square equals 1. If we put a circle in this area, and let the circle represent the probability of E, then the area outside of the circle represents probability of E prime. So, if I said the area of E is .35, then 1 minus .35 equals .65, and that is the area of E prime.

And as you just saw, we can use this equation 2 other ways. First, we can subtract probability of E prime from both sides to get the probability of E equals 1 minus the probability of E prime, or again from our original equation, we can subtract probability of E from both sides to get probability of E prime equals 1 minus probability of E. And these equations can come in handy.

For instance, lets say you polled 50 people and asked them what their favorite sport is. Using a frequency table, 29 said football, 11 said basketball, 6 said baseball, 3 said soccer, and 1 said tennis. Based on the data in this table, what is the probability randomly selecting someone who’s favorite sport is not football?

So, to answer this, we can use the equation probability of E prime equals 1 minus probability of E. In this equation, the event E is probability that football is a person’s favorite sport, which is 29 over 50, and E prime is football is not a person’s favorite sport.

So, probability of E prime equals 1 minus 29 over 50, doing some math we get 50 over 50 minus 29 over 50, which equals 21 over 50 which equals 0.42 or a 42% probability of selecting someone who’s favorite sport is not football.

In calculating probabilities, this equation can be very useful in many situations.

Timestamps
0:00 What Is The Complement Of An Event?
0:23 Rules For Complementary Events
0:57 Formulas For Complementary Events
1:45 Complementary Event Example Problem

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