Chapter 7: Probability

Lesson 5: Independent Events

Mrs. Parziale

Vocabulary:

• Independent events: the results of one event do not affect the results of the other event.– Events A and B are independent events IFF

P(AB)=P(A) • P(B)Events with replacement are considered to be

independent because later events do not remember what happed with earlier selections.

Independent Example

• Suppose that six socks are in a drawer:

We name them 1 2 3 4 1 2. If a first sock is blindly taken, put back, and then a second sock is taken, what is the probability that both are blue?

Show that P(AB)=P(A) • P(B)

Independent Events

• Event A = first sock is blue• Event B = second sock is blue

11 12 13 14 11 12 21 22 23 24 21 22 31 32 33 34 31 32 41 42 43 44 41 42 11 12 13 14 11 12 21 22 23 24 21 22

( )P A B

( )P A ( )P B

Vocabulary:

• Dependent events: the results of one event do affect the results of the other event.– Events A and B are dependent events when

P(AB)≠P(A) • P(B)

Dependent Example

• Suppose that six socks are in a drawer:

We name them 1 2 3 4 1 2. If a first sock is blindly taken, without replacement, and then a second sock is taken, what is the probability that both are blue?

Show that P(AB) ≠ P(A) • P(B)

Dependent Events

• Event A = first sock is blue• Event B = second sock is

blue( )P A B

( )P A ( )P B

12 13 14 11 12 21 23 24 21 22 31 32 34 31 32 41 42 43 41 42 11 12 13 14 12 21 22 23 24 21

11 12 13 14 11 12 21 22 23 24 21 22 31 32 33 34 31 32 41 42 43 44 41 42 11 12 13 14 11 12 21 22 23 24 21 22

Example 1: • The circular region around a fair spinner is

divided into six congruent sectors as pictured below. Consider spinning it twice. (Suppose the spinner cannot stop on the boundary lines.)

• Event A: The first spin stops on an even number.• Event B: The second spin stops on a multiple of

3.

6 1

5 2

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Are these dependent or independent events? a. Find and b. Find the sample space, and then find .

( )P A ( )P B

( )P A B

Example 2: A bag contains five marbles: three red and two blue (we will call

them R1, R2, R3, B1, and B2. Find the probability that if you draw two marbles, that they both are blue. (with replacement)

Let A be the event that the first marble is blue, and B be the event that the second marble is blue. a. Find and

b. Find the sample space, and then find .

( )P A ( )P B

( )P A B

r1,r1 r1,r2 r1,r3 r1,b1 r1,b2r2,r1 r2,r2 r2,r3 r2,b1 r2,b2r3,r1 r3,r2 r3,r3 r3,b1 r3,b2b1,r1 b1,r2 b1,r3 b1,b1 b1,b2b2,r1 b2,r2 b2,r3 b2,b1 b2,b2

Example 3: A bag contains five marbles: three red and two blue (we will call

them R1, R2, R3, B1, and B2. Find the probability that if you draw two marbles, that they both are blue. (without replacement)

Let A be the event that the first marble is blue, and B be the event that the second marble is blue. a. Find and

b. Find the sample space, and then find .

( )P A ( )P B

( )P A B

r1,r1 r1,r2 r1,r3 r1,b1 r1,b2r2,r1 r2,r2 r2,r3 r2,b1 r2,b2r3,r1 r3,r2 r3,r3 r3,b1 r3,b2b1,r1 b1,r2 b1,r3 b1,b1 b1,b2b2,r1 b2,r2 b2,r3 b2,b1 b2,b2

r1,r2 r1,r3 r1,b1 r1,b2r2,r1 r2,r3 r2,b1 r2,b2r3,r1 r3,r2 r3,b1 r3,b2b1,r1 b1,r2 b1,r3 b1,b2b2,r1 b2,r2 b2,r3 b2,b1

Example 4:

A fair coin is tossed 4 times. Let A = getting all heads B = getting all tails

1. Are A and B independent or dependent? Why or why not?

2. Find P(A) and P(B) and P(A B).

Two normal dice are tossed. Let C = the sum is 7 D = the first dies shows a 5.

Are C and D dependent or independent?

Find P(C) and P(D) and P(C D).

ClosureUsing the spinner from the first example, consider the

following two events when the spinner is spun twice:Event A: the first spin shows a number less than 3.Event B: the sum of the spins is less than 5.

Are these events independent or dependent?Find P(A) and P(B) and P(A B).

6 1

5 2

34