Venn Diagram Example

By Henry Mesa

In a population of people, during flu season, 25% of that population got the flu. It turns out that 80% of that population had been given the current flu shot (vaccine) for that period of time. It turns out that the chance of randomly selecting someone who had taken the flu shot or had gotten the flu is 92%.

1st Step – write the given probabilities using function notation. You need to be able to do this otherwise your chance of success in not good.

P(flu) = 0.25 P(flu shot) = 0.8 and the last one is very important because it ties the other two events together; P(flu OR flu shot) = 0.92.

In a population of people, during flu season, 25% of that population got the flu. It turns out that 80% of that population had been given the current flu shot (vaccine) for that period of time. It turns out that the chance of randomly selecting someone who had taken the flu shot or had gotten the flu is 92%.

P(flu) = 0.25 P(flu shot) = 0.8 P(flu OR flu shot) = 0.92.

1. What is the probability of getting selecting someone at random and it turns out that this person received a flu shot and also had the flu?

2nd Step – write down the question using function notation.

P(flu AND flu shot) = ?

3rd Step – if possible create graph depicting the answer to your question.

I do not know anything about the distribution of the numbers. Thus I will use a Venn Diagram as a substitute.

In a population of people, during flu season, 25% of that population got the flu. It turns out that 80% of that population had been given the current flu shot (vaccine) for that period of time. It turns out that the chance of randomly selecting someone who had taken the flu shot or had gotten the flu is 92%.

P(flu) = 0.25 P(flu shot) = 0.8 P(flu OR flu shot) = 0.92

1. What is the probability of getting selecting someone at random and it turns out that this person received a flu shot and also had the flu?

P(flu AND flu shot) = ?

Notice that the ovals are not proportional to their

probabilities.

In a population of people, during flu season, 25% of that population got the flu. It turns out that 80% of that population had been given the current flu shot (vaccine) for that period of time. It turns out that the chance of randomly selecting someone who had taken the flu shot or had gotten the flu is 92%.

P(flu) = 0.25 P(flu shot) = 0.8 P(flu OR flu shot) = 0.92

1. What is the probability of getting selecting someone at random and it turns out that this person received a flu shot and also had the flu?

P(flu AND flu shot) = 0.25 + 0.8 – 0.92

= 0.13

And here is the answer by using the pictures.

= + -

This leads to the formula P(A AND B) = P(A) + P(B) – P(A OR B) which says if you want to calculate the P(A AND B) and if you have the information P(A), P(B), P(A OR B) then use the formula to get the answer.

In a population of people, during flu season, 25% of that population got the flu. It turns out that 80% of that population had been given the current flu shot (vaccine) for that period of time. It turns out that the chance of randomly selecting someone who had taken the flu shot or had gotten the flu is 92%.

P(flu) = 0.25 P(flu shot) = 0.8 P(flu OR flu shot) = 0.92.

2. What is the probability of getting selecting someone at random and it turns out that this person received a flu shot and did not get the flu?

2nd Step – write down the question using function notation.

P(NOT flu AND flu shot) = ?

3rd Step – if possible create graph depicting the answer to your question.

I do not know anything about the distribution of the numbers. Thus I will use a Venn Diagram as a substitute.

In a population of people, during flu season, 25% of that population got the flu. It turns out that 80% of that population had been given the current flu shot (vaccine) for that period of time. It turns out that the chance of randomly selecting someone who had taken the flu shot or had gotten the flu is 70%.

P(flu) = 0.25 P(flu shot) = 0.8 P(flu OR flu shot) = 0.70.

2. What is the probability of getting selecting someone at random and it turns out that this person received a flu shot and did not get the flu?

P(NOT flu AND flu shot) = ?

no

no

no

Yes!

I am going to try an identify which region satisfies the probability statement.

In a population of people, during flu season, 25% of that population got the flu. It turns out that 80% of that population had been given the current flu shot (vaccine) for that period of time. It turns out that the chance of randomly selecting someone who had taken the flu shot or had gotten the flu is 92%.

P(flu) = 0.25 P(flu shot) = 0.8 P(flu OR flu shot) = 0.70, P(flu AND flu shot) = 0.13

2. What is the probability of getting selecting someone at random and it turns out that this person received a flu shot and did not get the flu?

P(NOT flu AND flu shot)

= -

= 0.8 – 0.13

= 0.67

P(NOT flu AND flu shot) = ?

In a population of people, during flu season, 25% of that population got the flu. It turns out that 80% of that population had been given the current flu shot (vaccine) for that period of time. It turns out that the chance of randomly selecting someone who had taken the flu shot or had gotten the flu is 92%.

P(flu) = 0.25 P(flu shot) = 0.8 P(flu OR flu shot) = 0.70, P(flu AND flu shot) = 0.13

3. What is the probability of getting selecting someone at random and it turns out that this person does not receive a flu shot and did not get the flu?

P(NOT flu AND NOT flu shot) = ?

no

no

Yes!

no

I am going to try an identify which region satisfies the probability statement.

In a population of people, during flu season, 25% of that population got the flu. It turns out that 80% of that population had been given the current flu shot (vaccine) for that period of time. It turns out that the chance of randomly selecting someone who had taken the flu shot or had gotten the flu is 92%.

P(flu) = 0.25 P(flu shot) = 0.8 P(flu OR flu shot) 0.92, P(flu AND flu shot) = 0.35

3. What is the probability of getting selecting someone at random and it turns out that this person does not receive a flu shot and did not get the flu?

P(NOT flu AND NOT flu shot)

= -

P(NOT flu AND NOT flu shot) = ?

= 1 – 0.92

= 0.08

In a population of people, during flu season, 25% of that population got the flu. It turns out that 80% of that population had been given the current flu shot (vaccine) for that period of time. It turns out that the chance of randomly selecting someone who had taken the flu shot or had gotten the flu is 92%.

P(flu) = 0.25 P(flu shot) = 0.8 P(flu OR flu shot) = 0.92, P(flu AND flu shot) = 0.13

4. Given that a person has received a flu shot, what is the probability that this person contracted the flu?

P(flu | flu shot) = ? P(A AND B)

= P(B)

P(A | B)

P(flu AND flu shot) =

P(flu shot)P(flu | flu shot)

0.13 =

0.8

= 0.1625

In a population of people, during flu season, 25% of that population got the flu. It turns out that 80% of that population had been given the current flu shot (vaccine) for that period of time. It turns out that the chance of randomly selecting someone who had taken the flu shot or had gotten the flu is 92%.

P(flu) = 0.25 P(flu shot) = 0.8 P(flu OR flu shot) = 0.92, P(flu AND flu shot) = 0.13

P(flu | flu shot) = 0.1625

4. Are the two events a person gets the flu, a person gets a flu shot independent?

P(A AND B) =

P(B)P(A | B) This is the general formula for P(A | B)

However, if event A is independent of event B then P(A | B) = P(A), that is the original probability P(A) does not change when we consider the same event, A, in a different sample space, B in this case. That is what we mean when we write P(A | B) which is the probability of event A given that we are only considering it in a sample space that only contains situations which meet event B criteria.

So for our question, P(flu | flu shot) = 0.1625, but P(flu) = 0.25. The two are not equal, P(flu | flu shot) P(flu), thus the two events are not independent.

A Consequence of Two Events Being Independent

P(A AND B) =

P(B)In general P(A | B) but if the two events are independent then P(A | B) = P(A).

Now, lets do some of algebra. When we have independence notice that

P(A AND B) =

P(B)P(A)

And if we solve for P(A AND B) we get the following equation:

P(A)P(B) = P(A AND B)

WARNING! This equation only works if events A and B are independent.

In a population of people, during flu season, 25% of that population got the flu. It turns out that 80% of that population had been given the current flu shot (vaccine) for that period of time. It turns out that the chance of randomly selecting someone who had taken the flu shot or had gotten the flu is 92%.

P(flu) = 0.25 P(flu shot) = 0.8 P(flu OR flu shot) = 0.92, P(flu AND flu shot) = 0.13

P(flu | flu shot) = 0.1625

4. Are the two events a person gets the flu, a person gets a flu shot independent?

P(A)P(B) = P(A AND B) is only true if the events are independent.

So for this case, P(flu)P(flu shot) = 0.25(0.8)

= 0.2 and this result does not agree with what we arrived at in the first problem that P(flu AND flu shot) = 0.35 so we arrive at the same conclusion; the two events are not independent.

Thus, if you want to see if two events are or are not independent, you need to use the formulas P(A | B) = P(A) or the formula P(A)P(B) = P(A AND B).

Practice Problems

After an election it turns out that 55% of eligible voters voted in the election. Of the eligible voters, 45% were registered Republicans. Also, 39% of eligible voters were Republicans who voted.

1st Step – write the given probabilities using function notation. You need to be able to do this otherwise your chance of success in not good. Do this yourself first. Even if you don’t get it correct better now than later.

P(voted) = 0.55 , P(Republican) = 0.45, P(Republican and voted) = 0.39. Notice that my entire sample space consist of eligible voters.

After an election it turns out that 55% of eligible voters voted in the election. Of the eligible voters, 45% were registered Republicans. Also, 39% of eligible voters were Republicans who voted.

P(voted) = 0.55 , P(Republican) = 0.45, P(Republican and voted) = 0.39. Notice that my entire sample space consist of eligible voters.

1. What is the probability of choosing someone at random from the eligible voters and that person turns out to be a Republican or to have voted in the election? Try this on your own first.

P(Republican OR voted)

2nd Step – write down the question using function notation.

After an election it turns out that 55% of eligible voters voted in the election. Of the eligible voters, 45% were registered Republicans. Also, 39% of eligible voters were Republicans who voted.

P(voted) = 0.55 , P(Republican) = 0.45, P(Republican and voted) = 0.39. Notice that my entire sample space consist of eligible voters.

1. What is the probability of choosing someone at random from the eligible voters and that person turns out to be a Republican or to have voted in the election? Try this on your own first.

P(Republican OR voted)

3rd Step – if possible create graph depicting the answer to your question.

After an election it turns out that 55% of eligible voters voted in the election. Of the eligible voters, 45% were registered Republicans. Also, 39% of eligible voters were Republicans who voted.

P(voted) = 0.55 , P(Republican) = 0.45, P(Republican and voted) = 0.39. Notice that my entire sample space consist of eligible voters.

1. What is the probability of choosing someone at random from the eligible voters and that person turns out to be a Republican or to have voted in the election? Try this on your own first.

P(Republican OR voted)

P(Republican OR voted) = 0.55 + 0.45 – 0.39

= 0.61

After an election it turns out that 55% of eligible voters voted in the election. Of the eligible voters, 45% were registered Republicans. Also, 39% of eligible voters were Republicans who voted.

P(voted) = 0.55 , P(Republican) = 0.45, P(Republican and voted) = 0.39. Notice that my entire sample space consist of eligible voters. P(Republican OR voted) = 0.61

2. What is the probability of choosing someone at random from the eligible voters, and that person turns out to be a Republican that did not vote? Try this on your own first.

P(Republican AND NOT voted) = 0.45 - 0.39

= 0.06

After an election it turns out that 55% of eligible voters voted in the election. Of the eligible voters, 45% were registered Republicans. Also, 39% of eligible voters were Republicans who voted.

P(voted) = 0.55 , P(Republican) = 0.45, P(Republican and voted) = 0.39. Notice that my entire sample space consist of eligible voters. P(Republican OR voted) = 0.61

3. What is the probability of choosing someone at random from the eligible voters and that person turns out to be a Republican or an eligible voter that did not vote? Try this on your own first.

P(Republican OR NOT voted) = 1 – (0.55 – 0.39)

= 0.84

After an election it turns out that 55% of eligible voters voted in the election. Of the eligible voters, 45% were registered Republicans. Also, 39% of eligible voters were Republicans who voted.

P(voted) = 0.55 , P(Republican) = 0.45, P(Republican and voted) = 0.39. Notice that my entire sample space consist of eligible voters. P(Republican OR voted) = 0.61

4. What is the probability of choosing someone at random from the eligible voters and that person turns out to not be a Republican and also this person was an eligible voter that did not vote? Try this on your own first.

P(NOT Republican AND NOT voted) = 1 – 0.61

= 0.39

After an election it turns out that 55% of eligible voters voted in the election. Of the eligible voters, 45% were registered Republicans. Also, 39% of eligible voters were Republicans who voted.

P(voted) = 0.55 , P(Republican) = 0.45, P(Republican and voted) = 0.39. Notice that my entire sample space consist of eligible voters. P(Republican OR voted) = 0.61

5. What is the probability of choosing someone at random from the eligible voters who is a Republican and have that person happens to have voted in the election. Try this on your own first.

P(voted | Republican) 0.39

= 0.45

= 0.8667

P(A AND B) =

P(B)P(A | B)

After an election it turns out that 55% of eligible voters voted in the election. Of the eligible voters, 45% were registered Republicans. Also, 39% of eligible voters were Republicans who voted.

P(voted) = 0.55 , P(Republican) = 0.45, P(Republican and voted) = 0.39. Notice that my entire sample space consist of eligible voters. P(Republican OR voted) = 0.61

6. What is the probability of choosing someone at random from the eligible voters who voted and this person turns out to be a Republican? Try this on your own first.

P(Republican | voted) 0.39

= 0.55

= 0.7091

P(A AND B) =

P(B)P(A | B)

After an election it turns out that 55% of eligible voters voted in the election. Of the eligible voters, 45% were registered Republicans. Also, 39% of eligible voters were Republicans who voted.

P(voted) = 0.55 , P(Republican) = 0.45, P(Republican and voted) = 0.39. Notice that my entire sample space consist of eligible voters. P(Republican OR voted) = 0.61

7. Are the events, “a person voted”, “ a person is a Republican” independent? A yes or no answer will not get any points in an examination.

P(Republican | voted) 0.39 =

0.55

= 0.7091

This does not equal P(Republican) = 0.45. While Republicans make up 45% of the eligible voter population they make up 70.91% of the people who voted. Thus, the two events are not independent.

This question could have been answered differently, by showing P(voted | Republican) does not equal P(voted), or P(Republican)P(voted) does not equal P(Republican AND voted)

After an election it turns out that 55% of eligible voters voted in the election. Of the eligible voters, 45% were registered Republicans. Also, 39% of eligible voters were Republicans who voted.

P(voted) = 0.55 , P(Republican) = 0.45, P(Republican and voted) = 0.39. Notice that my entire sample space consist of eligible voters. P(Republican OR voted) = 0.61

7. Are the events, “a person voted”, “ a person is a Republican” independent? A yes or no answer will not get any points in an examination. Another approach.

P(Republican and voted) = 0.39 while

P(voted)P(Republican) = 0.55(0.45)

= 0.2475

The formula P(A)P(B) = P(A and B) works when the events A, B are independent. Since we did not get equality this must mean the two events are not independent.

View this as often as you need.