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### Course: AP®︎/College Statistics>Unit 7

Lesson 2: Mutually exclusive events and unions of events

# Addition rule for probability (basic)

One hundred students were surveyed about their preference between dogs and cats. The following two-way table displays data for the sample of students who responded to the survey.
PreferenceMaleFemaleTOTAL
Prefers dogs$36$$20$$56$
Prefers cats$10$$26$$36$
No preference$2$$6$$8$
TOTAL$48$$52$$100$
problem 1
Find the probability that a randomly selected student prefers dogs.
$P\left(\text{prefers dogs}\right)=$

Problem 2
In this sample, are the events "prefers dogs" and "prefers cats" mutually exclusive?

problem 3
Find the probability that a randomly selected student prefers dogs or prefers cats.
$P\left(\text{prefers dogs or cats}\right)=$

problem 4
In this sample, are the events "prefers dogs" and "female" mutually exclusive?

problem 5
Find the probability that a randomly selected student prefers dogs or is female.
$P\left(\text{prefers dogs or is female}\right)=$

## Want to join the conversation?

• I am eternally confused by wording (perhaps being ESL?). In the question: In this sample, are the events "prefers dogs" and "prefers cats" mutually exclusive? If someone likes both cats and dog equally, and for that reason chooses no preference, then doesn't the answer become no, not mutually exclusive? How do you know what they actually mean vs reading questions with the wrong assumptions about what they are even asking? Thanks!!
• Mutually exclusive, simply means that, a person can't choose BOTH "prefer cats" and "prefer dogs" simultaneously. Hope this clarify your doubt.
• Why do I have to do homework if I can study at school? What is school for? Where is my freedom? I am willing to do my best at school but I also want to have time out of school.
• Hi, Great job on working hard, good job on planning some fun, but be sure that you do not do more work than you need to in order to understand each class, score as well as you would like to and accomplish your goals. After college, I realized I worked myself much harder than I needed to and didn't enjoy my life as much as I wished. Work hard. Sleep hard. Enjoy the journey! Life is NOT in the "when I finish college," "when I get a job," "after I move," "after I get married," "after I retire," "when I get more time" etc. Life is in the journey. Enjoy every step of the way!
• How do we tell when a question for probability is mutually exclusive?
• If two events are mutually exclusive they can't both happen at the same time.

If 𝐴 is the event that I drive a car,
𝐵 is the event that I ride a bike,
and 𝐶 is the event that I wear sunglasses,

then 𝐴 and 𝐵 are mutually exclusive, but 𝐴 and 𝐶 are not mutually exclusive.
• What exclusive means?
• Here, if two events are "mutually exclusive", then they can't both happen at the same time. If you flip one coin, it can be heads, or it can be tails, but it can't be both. Heads and tails are mutually exclusive.
• Are non-simplified answers acceptable or not?
• No, my answers was like 88/100, 56/100 etc. And system accept them. If your answer is correct, system does not need you to simplify it.
• I am extremely confused with Mutually Exclusive and Not Mutually Exclusive. Is there an easier way to remember and differentiate between them?
• If two events are mutually exclusive, then the two events both exclude each other (they have nothing in common). If they are not mutually exclusive, there is the possibility that an observation can be an outcome in both events.
• when do you need to subtract a and b
• You need to subtract "a and b" when you are solving for a problem with overlap.
For instance, in problem 5, a student could prefer dogs AND be female. When you calculate the probability that a student prefers dogs or is female, understand that each category (student that prefers dogs vs student that is female) contains members of the other category, as they are not mutually exclusive; some female students prefer dogs ('female students' includes some students who 'prefer dogs').
Because of this overlap, you can't just add the two probabilities together-- that would over-represent the female students who like dogs. So you subtract that overlap ("a and b") once from your result.
When A and B overlap: P(A or B) = P(A) + P(B) - P(A and B)
• I have a test soon, wish me luck.
• Good luck!
• There are 500 students in a high school senior class. Of these 500 students, 300 regularly wear a necklace to school, 200 regularly wear a ring, and 125 regularly wear a necklace and a ring. Using this information, answer each of the following questions.

The probability of of a student wears a ring or a necklace was 3/4. Can someone explain this to me, how it's less than one while all students wear either ring or a necklace ?
• 125 of the students wear both a necklace and a ring.

This means that 125 of the students who wear a necklace are also included in the group of students who wear a ring.

So, when saying that the number of students who wear a necklace or a ring is 200 + 300 = 500, we actually count those 125 students twice!

In order to make up for this mistake we must now subtract 125,
and so the correct number of students who wear a necklace or a ring is
500 − 125 = 375

Thus, the proportion of students who wear a necklace or a ring is
375∕500 = 3∕4

– – –

Also, by using the given information, we can construct the following two-way table:
 Ring No Ring TOTAL
Necklace 125 175 300
No Necklace 75 125 200
TOTAL 200 300 500

The number of students who wear...
...a necklace and a ring = 125
...a necklace and no ring = 175
...a ring and no necklace = 75

Thereby, the number of students who wear a necklace or a ring =
= 125 + 175 + 75 = 375