Laws of Probability

probability (moore), some laws of.doc

25-Sep-13

Some Laws of Probability

For a sample space, S, and events A and B, we have the following laws:

General “Addition Law”: P(A or B) = P(A) + P(B) P(AandB)

Special “Addition Law”, if (AandB) = , i.e. A and B have no outcomes in common, they are said to be mutually exclusive (in set theory A and B are called disjoint). This special case of the “addition law” is P(A or B) = P(AorB) = P(A) + P(B)

Complement Law: (note: I’ll write “the eventA will not occur” as “not-A”)
P(not-A) = 1 P(A)

Exercise 1:Consider the experiment of rolling a fair die. Given S = {1, 2, 3, 4, 5, 6},
A = {1, 3, 5}, B = {2, 3, 5}, and C = {1, 6} verify that the above laws by computing each of the probabilities in two ways:

1. Find the event, and directly compute the probability

a)not-A = ______P(not-A) = ______

b)not-C = ______P(not-C) = ______

c)AorB = ______P(AorB) = ______

d)AorC = ______P(AorC) = ______

e)BorC = ______P(BorC) = ______

2. Use the laws above (show how you used the law(s))

a)P(not-A) = ______

b)P(not-C) = ______

c)P(AorB) = ______

d)P(AorC) = ______

e)P(BorC) = ______

Exercise 2:Let P(A) = 0.3, P(B) = 0.6 and P(AandB) = 0.2, find each of the following:

a)P(not-A)

b)P(not-B)

c)P(AorB)

d)Draw a Venn diagram to illustrated the relationship between these events

e)Use the Venn diagram to find P(not-Aandnot-B) (the probability that neither A nor B occur)

f)Use the Venn diagram to find P(Aandnot-B) (the probability that A occurs but B does not).

Conditional Probability

We often want to describe the probability of one event occurring given that another event has already occurred. For example we might want to know the probability of getting an A in a class given that you get an A on the final exam. This probability will most likely not be the same as the probability of getting an A in the class given you got a C on the final. Another example, the probability that a randomly selected SCC student is over 5 feet 10 inches tall may be 0.20, whereas the probability that a student is over 5 feet 10 inches given that the student is male may be 0.35. We write the probability that B given A, P(B|A)

Example:A card is randomly selected from a standard 52 card deck. P(heart) = 0.25, but P(heart|red) = 0.5.

Exercise 3:Consider the experiment of rolling a fair die. Given S = {1, 2, 3, 4, 5, 6},
A = {1, 3, 5}, B = {2, 3, 5}, C = {1, 6} and D = {1, 2, 3, 4}, find:

a)P(B|A)

b)P(A|B)

c)P(C|A)

d)P(A|C)

e)P(B|C)

f)P(A|D)

Exercise 4:The following is a two-way table classifying 500 SCC students who have transferred by the type of college to which transferred and the category of their major.

Social
Science / Natural
Science / Human-
ities / Other /
Total
UC / 25 / 50 / 20 / 5 / 100
CSU / 60 / 80 / 100 / 60 / 300
Private / 10 / 20 / 20 / 10 / 60
Other state public / 5 / 0 / 30 / 5 / 40
Total / 100 / 150 / 170 / 80 / 500

Assuming that a student is selected at random find:

a)P(CSU transfer)

b)P(CSU natural science major)

c)P(not a humanities major)

d)P(natural science major | CSU)

e)P(CSU | natural science major)

For a sample space, S, and non-empty events A and B,

Note that from these we can derive the “multiplication laws”

P(AandB) = P(A)P(B|A) and also, P(AandB) = P(B)P(A|B)

Exercise 5:Let P(A) = 0.3, P(B) = 0.6 and P(AandB) = 0.2, find:

a)P(B|A)

b)P(A|B)

Independent Events

The concept of independence of events in a very important one in statistics. Intuitively, two events are independent if when one event occurs the probability that the other occurs is unchanged. For example, if the probability that someone is blood type A is 0.30 and the probability that someone is type A given that they are a woman is also 0.30 then the event of being type A and the event of being a woman are independent. One of the major questions that we try to answer statistically is whether or not two events (or variables) are independent.

Caution, students often confuse independent events and mutually exclusive events, they are quit different, in fact they are contrary; if two non-empty events are independent then they are not mutually exclusive and if they are mutually exclusive then they are not independent. Note that if A and B are mutually exclusive then P(B|A) = 0 and P(A|B) = 0.

If A and B are independent then we have an important special case of the “multiplication law”

P(AandB) = P(A)P(B), this can be generalized to more than two independent events.

Example:If A and B are independent and P(A) = 0.4 and P(B) = 0.5 then P(AandB) = 0.2.

Example:Assuming that left handedness and blood types are independent and that 12% of a population is left handed and 30% are type A then what percentage are left handed and type A? P(left handed and type A) = P(left handed)P(type A) = (.12)(.30)
= 0.036 so 3.6% of the population is left handed and type A.

Exercise 6:Which of the sets in exercise 3 are independent?

Exercise 7:A box contains 5 red and 3 white balls.

1.If two balls are drawn at random without replacement (i.e. a first ball is draw, not returned to the box and then a second ball is drawn), find the probability:

a) the first is red and the second is white

b) they are both red

c) they are both white

d) there is one of each color drawn

2.If two balls are drawn at random with replacement (i.e. a first ball is draw, it is returned to the box and then a ball is again drawn), find the probability:

a) the first is red and the second is white

b) they are both red

c) they are both white

d) there is one of each color drawn

Exercise 8:Given that P(A) = 0.2 and P(B) = 0.7

1.find P(A and B) given that

a) A and B are mutually exclusive

b)A and B are independent

2.find P(A or B) given that

a) A and B are mutually exclusive

b)A and B are independent

Answers to the exercises:

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probability (moore), some laws of.doc

25-Sep-13

1.a) 1/2

b) 2/3

c) 2/3

d) 2/3

e) 5/6

2.a) 0.7

b) 0.4

c) 0.7

e) 0.3

f) 0.1

3.a) 2/3

b) 2/3

c) 1/3

d) 1/2

e) 0

f) 1/2

4.a) 0.60

b) 0.16

c) 0.66

d) 4/15  0.27

e) 8/15  0.53

5.a) 2/3

b) 1/3

6.AC, AD

and BD

7.1.a) 15/56

b) 20/56 = 5/14

c) 6/56 = 3/28

d) 30/56 = 15/28

7.2.a) 15/64

b) 25/64

c) 9/64

d) 30/64 = 15/32

8.1a) 0.00

b) 0.14

8.2a) 0.90

b) 0.76

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