The Probability of A or B - The Probability of A and B probability_of_A_or_B_and_A and B.doc Update 9/16/10 (For Chapter 4 in Triola 11e)

Simple Probability of A: total

successes

fromchoosetoelementsofnumberTotal

AinelementsofNumberAP

#)( ==

Example: What is the probability of drawing an Ace (there are 4) in a single draw of a card from a deck of 52 cards?

Solution: 13

1

52

4)( ==AceP Note: All probabilities are between 0 and 1 inclusive. Sample space S = All possible simple outcomes.

The Probability of A or B (a Compound Probability) P(A or B)

To calculate P(A or B), we must first discuss:

Mutually Exclusive/Disjoint Events: Let A and B be events the do not overlap. Then A and B are mutually exclusive (or disjoint).

Mutually exclusive events cannot happen simultaneously.

Mutually Exclusive: When tossing a single die, you cannot get a 2 and a 5 at the same time. The events of getting a 2 and getting a 5

in a single toss of a die are mutually exclusive. We can see that P(2 or 5) = 2/6 since there are 2 successes out of 5 total possible

outcomes. Also, we notice that P(2 or 5) = P(2) + P(5) = 1/6 + 1/6 = 2/6.

NOT Mutually Exclusive: When tossing a single die, you can get an odd number {1, 3, 5} and a number greater than 2 {3, 4, 5, 6} at

the same time. Hence, the events of getting an odd number and a number greater than 2 in a single toss of a die are not mutually

exclusive. We see that P(1, 3, 5 or 3, 4, 5, 6) = P(1, 3, 4, 5, 6) = 5/6 since we dont count the 3 and 5 twice. Notice that we can get

the same result by adding the individual probabilities (as in the previous example) and then subtracting out the probability of the

overlap:

P(1, 3, 5 or 3, 4, 5, 6) = P(1, 3, 5) + P(3, 4, 5, 6) - P(3, 5) = 3/6 + 4/6 2/6 = 5/6

Another example of events that are not mutually exclusive (not disjoint) is getting an Ace and getting a Diamond when drawing a

single card. You can get both on a single draw so they can happen simultaneously and hence the events overlap and are not mutually

exclusive.

Another example of events that are mutually exclusive (disjoint) is getting a Democrat and getting a Republican when making a single

phone call. You cannot get both on a single call so they cant happen simultaneously and hence the events do not overlap and are

mutually exclusive/disjoint.

Also, a moderate can also be a Republican or a Democrat. But nobody is both a Republican and a Democrat . Hence the events of

being a moderate and a Democrat and the events of being a moderate and a Republican are not disjoint while the events of being a

Republican and a Democrat are disjoint.

When events A and B ARE disjoint, the probability of one or the other happening is the sum of the probabilities. That is

P(A or B) = P(A) + P(B).

Example: There are 20% Republicans (R) (the right-wing base), 25% Democrats (D) (the left-wing base) and 55% Independents (I) in

a large population. What is the probability of randomly calling a Republican or an Independent on a single phone call

Solution: The events ARE mutually exclusive.

P(R) = .20 and P(I) = .55.

Hence, P(R or I) = P(R) + P(I) = .20 + .55 = .75 or 75%.

When events A and B are NOT disjoint is continued on the next page-

p2 of The Probability of A or B - The Probability of A and B

When events A and B are NOT disjoint, the probability of one or the other happening is the sum of the probabilities minus the

probability of the overlap (intersection). That is

P(A or B) = P(A) + P(B) - P(A and B).

Addition Rule

Example: In a crowd of 200, there are 60 Republicans (R), 40 Tea Partiers (T) and 35 Tea Partiers that are also Republicans. What is

the probability of randomly calling a Republican or a Tea Partier in a single phone call

Solution: The events are NOT mutually exclusive.

P(R) = 60/200 and P(T) = 40/200 and P(R and T) = 35/200.

Hence, P(R or T) = P(R) + P(T) - P(R and T) = 60/200 + 40/200 35/200 = 65/200 = .325 or 32.5%.

Using the Informal Addition Rule: 60/200 + 5/200 (adjusted) = 65/200 = .325 or 32.5%.

The Probability of A and B (a Compound Probability) P(A and B)

To calculate P(A and B), we must first discuss: Independent and Dependent Events Let A and B be events such that the occurrence of one does not affect the probability of the

occurrence of the other. Then A and B are independent.

For example, when choosing an Ace and then a Ten from a deck when the Ace is replaced before choosing the Ten, the probability of

choosing the Ten 4/52 was not affected by choosing the Ace first (since the Ace was replaced). But, if the Ace was not replaced

before choosing the Ten, the probability of choosing the Ten 4/51 would change from what it would have been if the Ace would have

been replaced. Hence, the events are independent when choosing with replacement and dependent when choosing without

replacement. (With large populations, we always use sampling with replacement and hence, we treat the events as

independent.)

When events A and B are independent, the probability of A and B is the product of the probabilities. That is

P(A and B) = P(A)P(B).

Example: There are 20% Republicans (R) (the right-wing base), 40% Democrats (D) (the left-wing base) and 40% Independents (I) in

a large population. What is the probability of randomly calling a Republican and an Independent when making 2 phone calls? (With

large populations, we always use sampling with replacement.)

Solution: The events are independent when we use sampling with replacement.

P(R) = .20 and P(I) = .40.

Hence, P(R and I) = P(R)P(I) = .20(.40) = .08 or 8%.

When events A and B are NOT independent, the probability of A and B is the product of the probability of A times the

probability of B given that A has already occurred. That is

P(A and B) = P(A)P(B|A).

Multiplication Rule

Example: The probability of rain is .80. The probability of getting wet given that it rained is .20. Find P(rain and wet).

Solution: The events are dependent. Hence P(rain and wet) = P(rain)P(wet given that it rained) = .80(.20) = .16

For another Example using the Informal Multiplication Rule, see above (middle of page) when the Ace was NOT replaced.

Good News: In later chapters in this course, most if not all events

will be disjoint when doing ORs and independent when doing

ANDs. We will simply add or multiply the probabilities.

Informal Addition Rule: To find P(A or B), find the sum of the number

of ways event A can occur and the number of ways event B can occur,

adding in such a way that every outcome is counted only once. P(A or B)

is equal to that sum, divided by the total number of outcomes.

Informal Multiplication Rule: When finding the probability that event A occurs in one trial and event B occurs in

the next trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B

takes into account the previous occurrence of event A.