Combined Probability

Prob/Stats 3.1.3 Probability Notation

How can I express Probabilities?

Open Ended

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Reminder to take notes!

You may have noticed that there are many reoccurring words in probability problems like “and,” “or,” and “not.” Just like you have been using P() notation to express probabilities, there are symbols commonly used to represent these various events. Some of them are shown here in the context of a relative frequency table:

3-20: APPLES AND BANANAS

Let event “A” represent a randomly selected high school student ate an apple at lunch and event “B” represent the student ate a banana at lunch. The event "did NOT eat an apple" is then "A bar" and "did NOT eat a banana" is "B bar".

The right notation can save a lot of explaining.

Multiple Choice

For example, answer each of the next two questions in a complete sentence:  

$$P\left(B\right)=$$  

• The probability of randomly selecting a high school student who ate a banana at lunch is 0.65.

• The probability of randomly selecting a high school student who did not eat a banana at lunch is 0.65.

• The probability of randomly selecting a high school student who ate a banana at lunch is 0.20.

• The probability of randomly selecting a high school student who ate a banana at lunch is 0.45.

Multiple Choice

For example, answer each of the next two questions in a complete sentence:  

$$P\left(\overline{A}\right)=$$  

• The probability of randomly selecting a high school student who did not eat an apple at lunch is 0.25. 
  

• The probability of randomly selecting a high school student who ate an apple at lunch is 0.25. 
  

• The probability of randomly selecting a high school student who did not eat an apple at lunch is 0.70. 

• The probability of randomly selecting a high school student who ate an apple at lunch is 0.70. 

Take Notes!!

• The symbol  $\cup$  stands for the “union” and represents the “or” condition

• The symbol  $\cap$  stands for “intersection” and represents the “and” condition

• Often the symbol “ | ” is used to represent “given that.” P(G|H) represents the probability of G given that H has occurred.

Multiple Choice

Choose the correct notation and then find the probabilities associated with randomly selecting a high school student who:

Ate a banana and an apple at lunch.

• $$P\left(B\cap A\right)$$ = 0.20

• $$P\left(B\cup A\right)$$ = 0.25

• P(B|A) = 0.31

• $P\left(B\cap A\right)$ = 0.25

• $P\left(B\cup A\right)$ = 0.75

Multiple Choice

Choose the correct notation and then find the probabilities associated with randomly selecting a high school student who:

Ate a banana or did not eat an apple at lunch.

• $$P\left(B\cap\overline{A}\right)$$  = 0.45
  

• $$P\left(B\cup A\right)$$ = 0.90

• P (B | $$\overline{A}$$ ) = 0.64

• $$P\left(B\cup\overline{A}\right)$$  = 0.90
  

•  $P\left(B\cup\overline{A}\right)$  = 0.45
  

Multiple Choice

Choose the correct notation and then find the probabilities associated with randomly selecting a high school student who:

Did not eat a banana and ate an apple at lunch.

• $$P\left(\overline{B}\cap A\right)$$  = 0.10
  

•  $P\left(\overline{B}\cap A\right)$  = 0.33

• P ( $$\overline{B}$$  | A ) = 0.33

• $$P\left(\overline{B}\cup A\right)$$  = 0.10
  

•  $P\left(\overline{B}\cup A\right)$  = 0.33
  

Multiple Choice

Choose the correct notation and then find the probabilities associated with randomly selecting a high school student who:

Ate a banana given that they ate an apple at lunch.

• $$P\left(B\cap A\right)$$  = 0.31
  

•  $P\left(B\cap A\right)$  = 0.20

• P (B|A) = 0.67

• P (A|B) = 0.67
  

• P (B|A) = 0.31
  

Multiple Choice

Here is the apples and bananas table with the relative frequencies removed. Some of the relative frequencies have been replaced with their associated probability notations.

Which table has the other missing relative frequencies with the correct probability notation?

Multiple Select

3-21: In a recent survey of college freshman, 35% of students checked the box next to “exercise regularly,” 33% checked the box next to “eat five or more servings of fruits and vegetables a day,” and 57% checked the box next to “neither.” Use the appropriate probability notation in following questions.

Draw a two-way relative frequency table in your notes to represent this situation.

Then select the variables that are being compared for this table.

• Regular Exercise

• Eating at least 5 servings of vegetables per day

• The number of freshman students at college

• College Freshman Grades

Poll

Here is an example of what your table could look like. It is also okay for you to have vegetables along the top, or for you to have used different letters such as $$\overline{V}$$ instead of  $$\overline{Fv}$$ .

How does the table in your notes compare?  Select more than one if you would like.

• I wrote basically the same thing and I feel confident about this.

• I got most of the numbers correct.

• I feel okay about this topic but want more practice.

• I struggled with placing the numbers but understood the variables or vice-versa.

• I didn't know where to start.

Fill in the Blank

What is the joint probability of randomly selecting a surveyed freshman who exercises regularly and eats 5
 servings of fruits and vegetables each day?

$$P\left(E\cap Fv\right)$$  =
Round your answer to the nearest whole percent.

Fill in the Blank

What is the probability of selecting a freshman who exercises regularly or eats 5 servings of fruits and vegetables each day?

$$P\left(E\cup Fv\right)$$  =
Round your answer to the nearest whole percent.

Draw a Conditional Relative Frequency Table in your notes.

Is there an association between eating fruits and vegetables and exercise? Create a conditional relative frequency table to help answer this question.  

Poll

Check your work. How did you do? Select as many answers as you would like.

• I got this correct and I feel confidant.

• I missed a few things but now that I see the example, I feel confident.

• I forgot how to do this and I don't understand the example picture above.

• I would like more practice creating conditional relative frequency tables.

Multiple Choice

Is there an association between eating fruits and vegetables and exercise? Use our conditional relative frequency table to help answer this question.

• Yes, there appears to be a strong association between exercise and eating at least 5 servings of vegetables each day.

• Yes, but the association appears to be weak between exercise and eating at least 5 servings of vegetables each day.

• No, there does not appear to be an association between exercise and eating at least 5 servings of vegetables each day.

Fill in the Blank

3-22: Lately there have been a number of times when the sound quality of the news interviews on the school’s video station has been unfit to broadcast. One source of the sound problem might be that one of the microphones is not working well. Brendan collected the following data from the last broadcast season:

Among all of the interviews, what is the probability that a randomly selected interview will have an unacceptable sound quality? Give your answer rounded to 2 decimal places.

Fill in the Blank

3-22: continued

What about a good sound quality? Give your answer rounded to 2 decimal places.

3-22: continued

Is there an association between the sound quality and the microphone used? Make a table in your notes with the conditional relative frequencies for each microphone.

Open Ended

Check your work from the last slide. Then use the table to answer the question:

Should Brendan keep searching for the source of the sound problem or has he found it? Explain.

Now that you have finished today's lesson, be sure to review any answers you got incorrect, then go to today's folder in Schoology and you can begin the homework assignment / or complete it on Friday!