Representing Data for Finding Probabilities

•There are 35 students•20 take math•25 take science•15 take both

•Venn Diagram Contingency table

155 10

5 M ^M

S 15 10 25

^S 5 5 10

20 15 35

Conditional Probability

P(B|A) means the probability B happens, given that A happens•P(Science|Math) = 15/20•P(Math|Science) = 15/25

You can get this from either representation

155 10

5 M ^M

S 15 10 25

^S 5 5 10

20 15 35

Independence

Two events, A and B are independent if P(B|A)=P(B)

Example:

I roll a die and flip a coin. P(H|6)=P(H) because the number on the die does not affect my chance of getting heads. Heads and Getting a Six are independent events.

Independence

Two events, A and B are independent if P(B|A)=P(B)

Example:

I pick a card out of a regular 52 card deck. Then, I pick a second card without replacing the first. Are the events getting a red card and then getting a queen independent?

No, because removing a card from the deck changes my probabilities for the second draw. My denominator is now 51 instead of 52.

Independence

Two events, A and B are independent if P(B|A)=P(B)

Example:

I pick a card out of a regular 52 card deck. Then, I pick a second card after putting the first card back in the deck. Are the events getting a red card and then getting a queen independent?

Yes, because replacing the first card makes it as if I never drew it in the first place. All probabilities remain the same for the second draw

Multiplication Rule

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

If A and B are independent, this becomes:

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

Multiplication Rule

Experiment: Flip a Coin and Roll a Die.

H1

H2

H3

H4

H5

H6

T1

T2

T3

T4

T5

T6

H

T

What is the probability of getting a head and then an even number?

P(H and E)=P(H)*P(E)

=

4

1

6

3

2

1

P(H and E) =

4

1

12

3

Multiplication Rule

There are 3 blue marbles and 2 green in a box. What is the probability you draw a blue and then a green?

P(B) = 3/5 P(G|B) = 1/2

X

10

3)(

2

1

5

3)(

)|(*)()(

BandGP

BandGP

BGPBPBandGP

Multiplication Rule

There are 3 blue marbles and 2 green in a box.

B

G

3/5

2/5

B

G

B

G

2/4

2/4

1/4

3/4

3/10

3/10

3/10

1/10

P(B and G) = 3/10

P(G and G) = 1/10

P(G|B) = 2/4=1/2

P(G|G) = 1/4

Multiplication Rule

What is the probability that I draw a queen and then a Ten when drawing without replacement?P(Q) = 4/52P(Ten|Q) = 4/51P(Q and ten) =

What is the probability that I draw a queen and then a Ten when drawing with replacement?P(Q) = 4/52P(Ten|Q) = 4/52P(Q and ten) =

663

4

2652

16

51

4

52

4

169

1

2704

16

52

4

52

4

Multiplication Rule

Here are the results for 240 students taking an entrance exam for placement in upper mathematics at a high school:

80

Passed Female

8040

40Is a student’s passing status independent of gender?

If so,P(passed) =P(passed|female)

P(passed) =160/240 = 2/3

P(passed|female)=80/120=2/3

P(passed)=P(passed|female) Passing is independent of gender for these results