5.2 Addition Rule and Complememnts (AND FLASHCARDS WITH HW)

• Disjoint is also called ‘Mutually Exclusive’

• What does it mean when two events are disjoint?

  • They have no outcomes in common

    • Another word for “disjoint” is “mutually exclusive”

      • So, events can be called disjoint or mutually exclusive

• Venn Diagram Tool

  • Rectangle = Sample space

  • Each circle = An event

    • Use letters “E” and “F” represent each event.

      • Label it in the circle.

  • Here is an example of what a Probability Venn Diagram should look like

    • EX: Suppose we randomly select a chip from a bag where each chip in the bag is labeled 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Let E represent the event “choose a number less than or equal to 2,” and let F represent the event “choose a number greater than or equal to 8.” These events are disjoint as shown in the figure.

      

How to calculate a single event’s probability:

“Disjoint” means “mutually exclusive” and describes the relationship between events:

In probability, "disjoint" is synonymous with "mutually exclusive," which means that only one of the events can happen at a given time.

Disjoint (or mutually exclusive) events are those that cannot happen at the same time, or have different values— so, only one of the events can happen at a given time. Note, events may be considered disjoint if they have different valuess; the key definition of disjoint events is that they cannot happen simultaneously.

• Example: If you flip a coin, getting heads and getting tails are disjoint events because you can only get one result on a single flip. 

Key takeaway: If two events happen at the same time, regardless of their values being different, they are not considered disjoint events. 

• The Addition Rule is used to calculate the probability that either (or both) of 2 or moew events will happen. The addition rule has two formulas, one for mutually exclusive events and one for non-mutually exclusive events:

1. The Addition Rule for Mutually Exclusive (Disjoint) Events

These are for Events that cannot happen at the same time.

• Statement: “If two events, A and B, are disjoint (meaning they have no common outcomes/cannot occur at the same time), then the probability of either/both event A or event B happening is simply the sum of their individual probabilities”

• Two events:

  • If two events ( E and F) are disjoint, then the probability of either one of them is found by…

    • 

• Three or more events:

  • If (E, F, G, … ) are all disjoint, then the probability for all of them is found by….

    • 

  ---

• EXAMPLE Addition Rule for Mutually Exclusive (Disjoint) Events

  • The probability model below shows the distribution of the number of rooms in housing units in the United States.

• (a) Verify that this is a probability model.

• (b) What is the probability a randomly selected housing unit has two or three rooms?

• (c) What is the probability a randomly selected housing unit has one or two or three rooms? 

• Part (a) steps:

  • All probabilities are between 0 and 1 (inclusive)

    • This means they all add up to 1

      • 0.010 + 0.032 + … + 0.080 = 1

    • Minitab function to sum probabilities:

      • Calc > Column Statistics…

      • Make sure “sum” is selected

      • Make sure proper column is in “Input Variable”.

• Part (b) steps:

  • P(two or three) 

    = P(two) + P(three) 

    = 0.032 + 0.093

    = 0.125

• Part ( c ) steps:

  • P(one or two or three)  

    = P(one) + P(two) + P(three)

    = 0.010 + 0.032 + 0.093

    = 0.135

2. Addition Rule for the probability of Non-Disjointed (Non-Mutually Exclusive) Events.

• Used when events can happen separately and/or may occur at the same time (i.e.,events overlap is possible)

• calculated by adding the probabilities of event E and event F, then subtracting the probability of both events occurring simultaneously (the overlap between E and F)

• The subtraction of P(A and B) prevents double counting. 

EXAMPLE Identifying a Non-Disjointed Event

According to a center for disease​ control, the probability that a randomly selected person has hearing problems is 0.158. The probability that a randomly selected person has vision problems is 0.094. Can we compute the probability of randomly selecting a person who has hearing problems or vision problems by adding these​ probabilities? Why or why​ not?

• No, because hearing and vision problems are not mutually exclusive.​ So, some people have both hearing and vision problems. These people would be included twice in the probability.

  • Some people have both hearing and vision problems. This means that the events of having a hearing problem and having a vision problem are not disjoint​ (or not mutually​ exclusive). Remember, when two events are not disjoint the General Addition Rule must be used, where the probability is found by adding together the two given probabilities and subtracting the probability of a person having both problems. If the probability of a person having both problems is not subtracted​ out, then these people will be included twice in the probability.

How to find components of each parts above^^^

Where​ N(E) is the number of outcomes in E and​ N(S) is the number of outcomes in the sample space.

Probability of a Disjoint Event Using the Complement Rule

Complement of an Event (E^C ) is the subset of outcomes in the sample space that are NOT in the event.

• The Complement Rule

  • If E represents any event and E^C represents the complement of E, then

    • P(E^C) = 1 – P(E)

  • Key Knowledge

    • is the subset of outcomes in the sample space that are NOT in the event.

    • Is itself an event.

    • An event and its complement are always mutually exclusive (they cannot occur simultaneously) and exhaustive (one of the two events must occur). 

    • Calculating Probability: To find the probability of an event's complement, subtract the probability of the original event from 1. 

    • Example: If event A is "rolling a 6 on a standard dice," then the complement of A would be "rolling any number other than a 6".

    • Complement of Event = E^C

      

EXAMPLE Illustrating the Complement Rule

• According to the American Veterinary Medical Association, 31.6% of American households own a dog.

  • (a) What is the probability that a randomly selected household does not own a dog?

    • P(do not own a dog) = 1 − P(own a dog)

                           = 1 − 0.316

          = 0.684

EXAMPLE Illustrating the Complement Rule

• The data to the below represent the travel time to work for residents of Hartford County, CT.

(a) What is the probability a randomly selected resident has a travel time of 90 or more minutes?

• There are a total of 24,358 + 39,112 + … + 4,895 = 393,186 residents in Hartford County.

  • The probability a randomly selected resident will have a commute time of “90 or more minutes” is…

    • 

(b) Compute the probability that a randomly selected resident of Hartford County, CT will have a commute time less than 90 minutes.

• P(less than 90 minutes) = 1 − P(90 minutes or more)

  = 1 − 0.012 = 0.988