JT Sec2 Concepts of Probability

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EMa1/CSci- Prob.& Stats - 9 - 2007-2008

CONCEPTS OF PROBABILITY

2.1 Random Experiments, Sample Spaces and Events

Probability theory is concerned with real life situations where a person performs an experiment the

outcome of which may not be certain. Such an experiment is called a random experiment. Games

of change are classic examples of random experiments - the outcome of throwing a dice is not

known with certainty prior to the throw. Firing a rocket is an example of performing a randomexperiment – it could result in failure or success.

Associated with any random experiment is a set S of all possible outcomes – this set S is called the

sample space of the random experiment. Each outcome in a sample space S is called a sample

point. An event is a subset of a sample space and contains those sample points for which the eventis true.

Example 1

Three items are selected at random from a manufacturing process. Each item is inspected andclassified defective (D) or nondefective (N). The sample space providing the most information

would be:

S = {NNN, NDN, DNN, NND, DDN, DND, NDD, DDD}

Let B be the event – ‘the number of defectives is greater than 1’.

B = {DDN, DND, NDD, DDD}

Example 2

An electronic component is placed on test and we are interested in the time to failure of the

component (in hours).

The sample space S = {t : 0 ≤ t < ∞}Let A be the event ‘the component fails before 5 hours’

A = {t : 0 ≤ t < 5}

Example 3

An experiment consists of tipping a coin and then flipping it a second time if a head occurs. If a tail

occurs on the first flip, then a dice is tossed.

The sample space S = {(H,H), (H,T), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}

A is the event ‘a number less than 4 occurred on the dice’,

A =

2.2 Operations with Events: Intersection, Union, Complement

We now consider operations with events that will result in the formation of new events – these new

events will be subsets of the same sample space as the given events.

Definitions

The Intersection of two events A and B, denoted by the symbol A∩B, is the event containing all

the sample points that are common to A and B.

The Union of two event A and B, denoted by the symbol A∪B, is the event containing all thesample points that belong to A or B or both.




The Complement of an event A with respect to a sample space S is the set of all sample points of S

that are not in A. The complement of A is denoted by A’.

Example 4

An electronic component is placed on test, let A be the event ‘the component fails before 5 hours’and B the event ‘the component fails before 10 hours’.

Not the sample space S = {t : 0 ≤ t < ∞}

A∩B = {t : 0 ≤ t < 5}

A∪B = {t : 0 ≤ t < 10}

A’ = {t : 5 ≤ t < ∞}, B’ = {t : 10 ≤ t < ∞}.

2.3 Mutually Exclusive Events

In certain statistical experiments we may define two events A and B that cannot occur

simultaneously – the events A and B are then said to be mutually exclusive and have no sample points in common ie A∩B = φ.

Example 5

Suppose a dice is tossed, let A be the event ‘an even number turns up’ and B is the event ‘an odd’

number turns up!

Then A = {2, 4, 6} B = {1, 3, 5}

and A∩B = φ hence A and B are mutually exclusive events.

Example 6

In the series circuit

We assume that R 1, R 2 can be in two possible states 1 or 0 denoting operative or defective.

The sample space S for the circuit is S = {(1, 1), (0, 1), (1, 0), (0, 0)}

If E1 is the event ‘the entire circuit is operative’ then E1 = {(1,1)} and E2 is the event ‘at least one of

R 1, R 2 is operative’ then E2 = {(1, 1), (0, 1), (1, 0)}.

We notice that E1 and E2 are not mutually exclusive since the sample point (1, 1) belongs to both

events.

R 2 R 1




2.4 Probability of an Event

The statistician is concerned with drawing conclusions from random experiments and, for these

conclusions to be reasonably accurate, an understanding of probability theory is essential.

Historically, the concept of the probability of an event has been explained by one of three methods

– the relative frequency method, the equally likely outcome method and the subjective ‘degree of belief’ method. The results obtained by these methods have led to the formation of an axiomaticdefinition of probability which avoids many of the pitfalls inherent in these historical methods.

The mathematical theory of probability for finite sample spaces provides a set of numbers called

weights, ranging from zero to 1, which provide a means of evaluating the likelihood of occurrence

of event resulting from a statistical experiment. To every point in the sample space we assign a

weight such that the sum of all the weights is 1. If we have a reason to believe that a certain sample

point is quite likely to occur when the experiment is conducted, the weight is assigned should be

close to 1. On the other hand, a weight closer to zero is assigned to a sample point that is unlikely

to occur. In many experiments, such as tossing a balanced coin or dice, all the sample points have

the same chance of occurring and are assigned equal weights.

Definition

The probability of any event A is the sum of the weights of all sample points in A. If we denote the

probability of event A by P(A) then

0 ≤ P(A) ≤ 1, P(φ) = 0, P(S) = 1

Example 7

A dice is loaded in such a way that an even number is twice as likely to occur as an odd number. If

E is the event that a number less than 4 occurs on a single toss of the dice, find P(E).

2.5 Equally likely outcomes

If the experiment is of such a nature that we can assume equal weights for the sample points of S,

then the probability of any event A is the ratio of the number of elements in A to the number of

elements in S.

Example 8

A bag contains 5 tags marked with the integers 1 through 5. Two tags are drawn at random, the first

tag being replaced before the second is drawn. Find the probability of the events A = {both tagsdrawn have the same numbers}, B = {second number drawn is strictly greater than the first number

drawn}.




Now the sample space contains the 25 ordered pairs (i, j), 1 ≤ i ≤ 5, 1 ≤ j ≤ 5 where the first number

of the pair indicates the first number drawn. Each ordered pair is equally likely to occur with

probability25

1.

Now A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)} and P(A) =25

5

And B = so P(B) =25

10

Comment

If the weights cannot be assumed equal in an experiment then they must be assigned on the basis of

prior knowledge or experimental evidence. For example if a coin is not balanced, we would

estimate the two weights for the outcomes head and tail by tossing the coin a large number of times

and recording the outcomes. The true weights would be the fractions of heads and tails that occur in the long run – this method of arriving at the weights is known as the relative frequency definition

of probability.

2.6 Probability Laws

Often it is necessary to calculate the probability of some event for known probabilities of other

events – we find it useful to derive probability laws for the union of two events and the complement

of an event.

Theorem: If A and B are any two events in a sample space S

P(A∪B) = P(A) + P(B) – P(A∩B)

Proof .

Referring to the Venn diagrams we notice that P(A∪B) is the sum of the weights of the sample

points in A∪B. Now P(A) + P(B) is the sum of all the weights in A plus the sum of all the weights

in B. We have added the weights in A∩B twice.

Hence P(A∪B) = P(A) + P(B) – P(A∩B)

Corollary 1

If A and B are mutually exclusive ie P(A∩B) = 0

then P(A∪B) = P(A) + P(B)

Example 9

Calculate the probability of getting a total of 7 or 11 when a pair of dice is tossed. Let A be the

event that 7 occurs then A = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}. Let B be the event that 11occurs then B = {(5, 6), (6, 5)}. The sample space S has 36 equally likely sample points.




So P(A) =36

6P(B) =

36

2P(A∩B) = 0

Then P(A∪B) = P(A) + P(B) since A and B are mutually exclusive =36

8.

Theorem:

If A and A′ are complementary events then P(A′) = 1 – P(A).

Proof :

Since A = A∪A′ and A∩A′ = φ

P(S) = P(A) + P(A′)

But P(S) = 1 so P(A′) = 1 – P(A)




Question Sheet 2

1.

An experiment involves tossing a pair of dice and recording the numbers that come up

(a) List the elements of the sample space S.

(b) List the elements of the event A that the sum is less than 5.(c) List the elements of the event B that a 6 occurs on at least one dice.

(d) List the elements of the event C that a 2 comes up on the first dice.

2.

Let the sample space S consist of all real numbers and define the events

}0:{C},21:{B},0:{A <=<<!=>= x x x x x x

(a) Describe and sketch on a number line the following events:-

A∪C, A∩C, A′, B∩C, A∩B′, A∪B

(b) Express the following events in terms of A, B and C:-

}.1:{},0:{},20:{ !"===<<= x x F x x E x x D

3.

An assembly of electronic equipment consists of 3 components arranged in a series-parallel circuit

as follows

Each component is either operative or fails under load. The entire assembly fails only if the path

from A to B is broken. Let the sample space S consist of the eight possible combinations of operative or inoperative components. Let E1 be the event ‘the assembly is operative’; let E2 be the

events ‘R 2 has failed but the assembly is operative’; let E3 be the event ‘R 3 has failed but the

assembly is operative’.

(a) List the sample points of S, E1, E2 and E3.

(b) Investigate whether E1, E2 and E3 are mutually exclusive.

4.

A coin is tossed until a head is obtained. Describe the appropriate sample space. If each point in

the sample space requiring n tosses has probability

n

!!

"

#

$$

%

&

2

1what is the probability that the first head

is obtained on an even-numbered toss?

5.

A fair coin is tossed 5 times and each outcome is equally likely. What is the probability of obtaining a sequence of at least three consecutive heads?

6.

If A and B are mutually exclusive events and P(A) = 0.4, P(B) = 0.5 calculate P(A∪B), P(A′),

P(A′∩B).

R 2

R 3

R 1 A B




7.

A coin is tossed and a dice is rolled. Calculate the probabilities of

(a) A tail and a 5.

(b) A head and an even number.

8.What is the probability of rolling two dice to obtain the sum, seven, and/or the number three on at

least one dice?

Answers to Sheet 2

1. (a) }61,61:),{(S !!!!= y x y x , i.e. 36 ordered pairs.

(b) A = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (3, 1)}

(c) B = {(1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}

(d) C = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)}

2. (a) A∪C = R / {0} R denotes the set of real numbers.

A∩C = φ

A′ = }0:{ !<"# x x B∩C = }01:{ <<! x x

A∩B′ = !<" x x 2:{ }, A∪B = }1:{ !<<" x x

(b) D = A∩B, E = A′∩C′, F = C∩B′

3. (a) Let 0 denote failure, 1 denote operative.

S = {(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0), (1, 1, 1)}

E1 = {(1, 1, 1), (1, 1, 0), (1, 0 1)} E2 = {(1, 0, 1)} E3 = {(1, 1, 0)}

(b) E2 and E3 are mutually exclusive.

4. P (first head on an even numbered toss) =3

1.

5.4

1

6. P(A∪B) = 0.9, P(A′) = 0.6, P(A′∩B) = 0.5

7. (a)12

1

(b)4

1

8.12

5

JT Sec2 Concepts of Probability

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