2.5. Formula for classical probability

Classical probability uses sample spaces to determine the numerical probability that an event will happen.

Classical probability assumes that all outcomes in the sample space are equally likely to occur. For example, when a single die is rolled, each outcome has the same probability of occurring. Since there are six outcomes, each outcome has a probability of 1/6. When a card is selected from an ordinary deck of 52 cards, we assume that the deck has been shuffled, and each card has the same probability of being selected. In this case, it is 1/52.

Definition:

If the sample space S contains n equally likely basic outcomes and the event A consists of m of these outcomes , then

In words, “The probability of event A equals to number of basic outcomes in A, divided by the total number of outcomes in the sample space”.

We can write definition above as .

Example:

For a card drawn from an ordinary deck find the probability of getting a queen

Solution:

Let A- be an event getting a queen. Since there are four queens then . Hence, .

Example:

Find the probability of obtaining an even number in one roll of a die.

Solution:

In this experiment S= {1, 2, 3, 4, 5, 6}.Let A- be an event that an even number is observed on the die. Event A has three outcomes: 2, 4, and 6.

If any one of these three numbers is obtained, event A is said to occur. Hence,

2.6. Consequences of the postulates

1. Let A and B be mutually exclusive events. Then the probability of their union is the sum of their individual probabilities;

that is

More generally, if are mutually exclusive events, then

2. If are collectively exhaustive events, then the probability of their union is .

Since the events are collectively exhaustive, their union is the whole sample space S and .

Example:

A drawer contains three pairs of red socks, two pairs of black socks and four pairs of brown socks. If a person in a dark room selects a pair of socks, find probability that the pair will be either black or brown. (Note: The socks are folded together in matching pairs).

Solution:

Let us define the following events

A= the selected socks are black

B= the selected socks are brown.

Since there are nine pairs of socks,

P (black) =P (A) = ; P (brown) =P (B) =

P(black or brown)= .

Example:

A day of the week is selected at random. Find the probability that it is a weekend day.

Solution:

Let

A= the selected day is Saturday

B= the selected day is Sunday

P(A)= ; P(B)= and

Exercises

1. B box contains three red and five blue balls. Define the sample space for experiment of recording the colours of three balls that are drawn from the box one by one, with replacement.

2. Define a sample space for the experiment of putting three different books on a shelf in random order. If two of these three books are a two-volume dictionary, describe the event that these volumes stand in increasing order side by side (i.e., volume I precedes volume II).

3. A simple card is drawn from an ordinary pack of playing cards. What is the probability that the card is

a) An ace b) A five

c) A red card d) A club

4. There are 15 slips of paper in a hat, numbered from 1 to 15. If one of slip is drawn at random, find the probability that

a) The number drawn is 5

b) The number drawn is even

c) The number drawn is odd

d) The number drawn is divisible by 3

5. Two events, A and B, are mutually exclusive: and .

Find the probability that

a) Either A or B will occur

b) Both A and B will occur

c) Neither A nor B will occur

6. The manager of a furniture store sells from zero to four sofas each week. Based on past experience, the following probabilities are assigned to sales of zero, one, two, three, or four sofas:

P (0) =0.08

P (1) =0.18

P (2) =0.32

P (3) =0.30

P (4) =0.12

1.00

a) Are these valid probability assignments? Why or why not?

b) Let A be the event that two or fewer are sold in one week. Find P (A).

c) Let B be the event that four or more are sold in one week. Find P (B).

d) Are A and B mutually exclusive? Find and .

7. Bektur, Janat, and Linar are the finalists in the spelling contest of a local school. The winner and the first runner-up will be sent to a city-wide competition.

a) List the sample space of concerning the out comes of the local contest.

b) Give the composition of each of the following events

A=Linar wins the local contest

B= Bektur does not go to the city-wide contest.

8. In a large department store, there are two managers, four department heads, 16 clerks, and four stokers. If a person selected at random, find the probability that the person is either a clerk or a manager.

9. On a small college campus, there are five English professors, four mathematics professors, two science professors, three psychology professors, and three history professors. If a professor is selected at random, find the probability that the professor is the following

a) An English or psychology professor.

b) A mathematics or science professor.

c) A history, science, or mathematics professor.

d) An English, mathematics, or history professor.

10. A hospital has monitored the length of time a patient spends in a hospital. The probability for numbers of days a patient spends in the hospital, are shown in following table:

Number of days	0	1-3	4-6	7-9	10-12	More than 12
Probability	0.14	0.39	0.23	0.15	0.06	0.03

Let A be the event “The patient spends at least one days in the hospital”, and B be the event “The patient spends less than10 days in the hospital”.

a) Find the probability of event A

b) Find the probability of event B.

c) Find the probability of the complement of A.

d) Find the probability of the union of A and B.

e) Find the probability of the intersection of A and B.

f) Are A and B mutually exclusive events?

g) Are A and B collectively exhaustive events?

Answers

1. { RRR, RRB, RBR, RBB, BRR, BRB, BBR,BBB};2.

; ; 3. a) ; b) ; c) ;

d) ; 4. a) ; b) ;c) ; d) ; 5. a) ; b) 0; c) ; 6. a) Yes;

b) 0.58; c) 0.12; d) Yes; ; ; 7. a) {BJ, BL, JB, JL,LB, LJ}; b) A={ LB, LJ}; B={ JL, LJ}; 8. ; 9. a) ; b) ;

c) ; d) ; 10. a) 0.86; b) 0.91; c) 0.14; d) 1; e) 0.77; f) no, because ; g) Yes, because .