Probability of Two Dependent Events: If two events, A and B, are dependent, then the probability of both events occurring is…

P(A and B) = P(A) * P(B following A)

13.4 Probabilityof Compound Events

Probability of Two Independent Events: If two events, A and B, are both independent, then the probability of both events occurring is…

P(A and B) = P(A)*P(B)

Example of Independent Events: What is the probability of rolling a 4 on a die 3 times in a row?

13.4 Probabilityof Compound Events

P(4) = P(three 4s) =

Roll 1 Roll 2 Roll 3

P(4) * P(4) * P(4)

=

≈ or 0.423%

Example of Dependent Events: What is the probability of drawing 6 hearts from a deck of cards without replacement?

13.4 Probabilityof Compound Events

P(six hearts) =

Draw 1 Draw 2 Draw 3

* * *

=

≈ or 0.008429%

Draw 4 Draw 5 Draw 6

* * 84 7

Example: Suppose the odds of the Sixers beating the Kings in Basketball was 5 : 2. What is the probability of the Sixers beating the Kings 4 times in a row?

13.4 Probabilityof Compound Events

Game 1 Game 2 Game 3

* * *

Odds = 𝑃 (𝑆𝑖𝑥𝑒𝑟𝑠𝑤𝑖𝑛)𝑃 (𝐾𝑖𝑛𝑔𝑠𝑤𝑖𝑛)

=

5727

Game 4

P(4 wins) =

P(Sixers win)

= = ≈

Example: A particular bag of marbles contains 4 red, 6 green, 2 blue, and 5 white marbles. What is the probability of picking a red, white, and blue marble, in that order?

What would the probability be with replacement?

13.4 Probabilityof Compound Events

P(r,w,b) =

Pick 1 Pick 2 Pick 3

* * P(r,w,b) = ≈ .0098

P(r,w,b) =

Pick 1 Pick 2 Pick 3

* * P(r,w,b) = ≈ .0081

13.4 Probability of MutuallyExclusive Events and Inclusive Events

Probability of Mutually Exclusive Events: If two events, A and B, are mutually exclusive, then the probability that either A OR B occurs is…

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

Mutually Exclusive Events: If two events, A and B, are mutually exclusive, then that means that if A occurs, than B cannot, and vice versa.

Probability of Inclusive Events: If two events, A and B, are inclusive, then the probability that either A or B occurs is…

P(A or B) = P(A) + P(B) – P(A and B)

Inclusive Events: If two events, A and B, are inclusive, then that means that if A occurs, B could also occur, and vice versa.

13.4 Probability of MutuallyExclusive Events and Inclusive Events

Example: A particular bag of marbles contains 4 red, 6 green, 2 blue, and 5 white marbles. If 3 marbles are picked, what is the probability of picking all reds or all greens?

13.4 Probability of MutuallyExclusive Events and Inclusive Events

P(red or greens) = P(red) + P(green)

=

=

= = ≈

mutually exclusive event

¿𝐶(4,3 )

𝐶(17 , 3)

¿4

680+

20680

= = ≈

+𝐶(6,3 )

𝐶(17,3)

Example: Slips of paper numbered 1 to 15 are placed in a box. A slip of paper is drawn at random. What is the probability that the number picked is either a multiple of 5 or an odd number?

13.4 Probability of MutuallyExclusive Events and Inclusive Events

inclusive event

P(mult of 5 or odd) = P(mult of 5) + P(odd) – P(5 and odd)

=

= ≈ =

Example: Two cards are picked out of a standard deck. What is the probability of both cards being either face cards or clubs?

13.4 Probability of MutuallyExclusive Events and Inclusive Events

inclusive event

P(face or clubs)

= =

=

=

≈

= P(face) + P(club) – P(face and club)

+[1352

∗ 1251 ]−[ 3

52∗ 2

51 ]

Example: 4 coins are tossed. What is the probability of obtaining 2 heads or 1 tail?

13.4 Probability of MutuallyExclusive Events and Inclusive Events

P(2 heads or 1 tail )

=

= = =

mutually exclusive event

=

= P(2 heads) + P(1 tail)

In a particular group of hospital patients, the probability of having high blood pressure is , the probability of having arteriosclerosis is , and the probability of having both is

a) Determine whether the events are mutually exclusive or mutually inclusive.

b) What is the probability that a patient in this group has either high blood pressure or arteriosclerosis?

13.4 Probability of MutuallyExclusive Events and Inclusive Events

Probability of Two Independent Events: P(A and B) = P(A)*P(B)

Probability of Two Dependent Events:

P(A and B) = P(A) * P(B following A)

Probability of Mutually Exclusive Events: P(A or B) = P(A) + P(B)

Probability of Inclusive Events: P(A or B) = P(A) + P(B) – P(A and B)