"The most important questions of life are, for themost part, really problems of probability."

Pierre-Simon Laplace (1749 - 1827)

6.3 MORE ABOUT PROBABILITY (Pages 341 - 355)

OVERVIEW: There are basic laws thatgovern wise and efficient uses of probability. An understanding ofthese laws is important for those who wish to utilize statistics inpractical and meaningful ways..

For any two events A and B,

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

If A and B are disjoint, then Prob(A and B) =0.

Prob(B|A) = prob(A and B)/prob(A).

Here are some illustrations of these probabilitylaws. Consider rolling two dice and calculating the sum on the upfaces. The sums are shown in the following table:

RED DIE

GREEN

DIE

1

2

3

4

5

6

1

2

3

4

5

6

7

2

3

4

5

6

7

8

3

4

5

6

7

8

9

4

5

6

7

8

9

10

5

6

7

8

9

10

11

6

7

8

9

10

11

12

Prob(sum=7 OR sum = 11) = 6/36 + 2/36 =8/36 = 22.22%.

Prob(sum=7 AND sum = 11) = 0.

Prob(sum=7 OR at least one die shows a 5) =6/36 + 11/36 - 2/36 = 15/36 = 41.67%.

Prob(faces show same number OR sum>9) = 6/36 + 6/36 -2/36 = 10/36 = 27.78%.

Prob(sum=6|one die shows a 4) = Prob(sum=6AND one dieshows a 4)/Prob(one die shows a 4) = (2/36)/(11/36) = 2/11 =18.18%.

Prob(sum is even|sum>9) = Prob(sum is evenANDsum>9)/Prob(sum>9) = (4/36)/(6/36) = 4/6 = 66.67%.

The probability formulas are useful, but oftentimes they are not needed if sample spaces are small and one usessome good old common sense. Here are some simple, yet sophisticated,examples involving conditional probability.

Consider a family with two children. There arefour possible boy/girl combinations, all equally likely.

 OLDEST YOUNGEST G G G B B G B B

Prob(at least one child is a girl) = 3/4.

Prob(two girls) = 1/4.

Prob(two girls|one child is a girl) = 1/3.Sample space is {(G,G), (G,B), (B,G)}.

Prob(two girls|oldest child is a girl) = 1/2.Sample space is {(G,G), (G,B)}.

Prob(exactly one boy|one child is a girl) =2/3.

Prob(exactly one boy|youngest child is a girl) =1/2.

OK, now suppose that a family unknown to you hastwo children. Here are two questions relating to this family.
...
(1) If one of the children is a boy,what is the probability that the other child is aboy?
...
(2) If the oldest child is a boy, whatis the probability that the other child is a boy?
Many intelligent people think that 50% is the answer to both (1) and(2). You should be able to show that this is not the case.