# Theoretical Probability

Definition of Theoretical Probability:

Let a random experiment produce only finite number of mutually exclusive and equally likely outcomes. Then the probability of an event E is defined as

Number of favorable outcomes
P(E) =     Total number of possible outcome

The formula for finding the theoretical probability of an event is

Number of favorable outcomes
P(E) =     Total number of possible outcome

Theoretical probability is also known as Classical or A Priori probability.

To find the theoretical probability of an event we need to follow the above explanation.

Problems based on Theoretical Probability

1. A fair coin is tossed 450 times and the outcomes were noted as: Head = 250, Tail = 200

Find the probability of the coin showing up

(ii) a tail.

Solution:

Number of times coin is tossed = 450

Number of tails = 200

(i) Probability of getting a head

Number of favorable outcomes
P(H) =     Total number of possible outcome

= 250/400
= 5/8

(ii) Probability of getting a tail

Number of favorable outcomes
P(T) =     Total number of possible outcome

= 200/400
= 1/2

2. In a cricket match the Sachin hit a boundary 5 times out of 30 balls he plays. Find the probability that he

(i) hit a boundary

(ii) do not hit a boundary.

Solution:

Total number of balls Sachin played = 30

Number of boundary hit = 5

Number of times he did not hit a boundary = 30 - 5 = 25

(i) Probability that he hit a boundary

Number of favorable outcomes
P(A) =     Total number of possible outcome

= 5/30
=1/6

(ii) Probability that he did not hit a boundary

Number of favorable outcomes
P(B) =     Total number of possible outcome

= 25/30
= 5/6

3. The record of weather stations report shows that out of the past 95 consecutive days, its weather forecast was correct 65 times. Find the probability that on a given day:

(i) it was correct

(ii) it was not correct.

Solution:

Total number of days = 95

Number of correct weather forecast = 65

Number of not correct weather forecast = 95 - 65 = 30

(i) Probability of ‘it was correct forecast’

Number of favorable outcomes
P(X) =     Total number of possible outcome

= 65/95
= 13/19

(ii) Probability of ‘it was not correct forecast’

Number of favorable outcomes
P(Y) =     Total number of possible outcome

= 30/95
= 6/19

4. In a society 1000 families with 2 children were selected and the following data was recorded

Find the probability of a family, having:

(i) 1 boy

(ii) 2 boys

(iii) no boy.

Solution:

According to the given table;

Total number of families = 333 + 392 + 275 = 1000

Number of families having 0 boy = 333

Number of families having 1 boy = 392

Number of families having 2 boys = 275

(i) Probability of having ‘1 boy’

Number of favorable outcomes
P(X) =     Total number of possible outcome

= 392/1000
= 49/125

(ii) Probability of having ‘2 boys’

Number of favorable outcomes
P(Y) =     Total number of possible outcome

= 275/1000
= 11/40

(iii) Probability of having ‘no boy’

Number of favorable outcomes
P(Z) =     Total number of possible outcome

= 333/1000

More solved examples on theoretical probability:

5. Two fair coins are tossed 225 times simultaneously and their outcomes are noted as:

(i) Two tails = 65,

(ii) One tail = 110 and

(iii) No tail = 50

Find the probability of occurrence of each of these events.

Solution:

Total number of times two fair coins are tossed = 225

Number of times two tails occur = 65

Number of times one tail occur = 110

Number of times no tail occur = 50

(i) Probability of occurrence of ‘two tails’

Number of favorable outcomes
P(X) =     Total number of possible outcome

= 65/225
= 13/45

(ii) Probability of occurrence of ‘one tail’

Number of favorable outcomes
P(Y) =     Total number of possible outcome

= 110/225
= 22/45

(iii) Probability of occurrence of ‘no tail’

Number of favorable outcomes
P(Z) =     Total number of possible outcome

= 50/225
= 2/9

6. A die is thrown randomly four hundred fifty times. The frequencies of outcomes 1, 2, 3, 4, 5 and 6 were noted as given in the following table:

Find the probability of the occurrence of the event

(i) 4

(ii) a number < 4

(iii) a number > 4

(iv) a prime number

(v) a number < 7

(vi) a number > 6

Solution:

Total number of times a die is thrown randomly = 450

(i) Number of occurrence of a number 4 = 75

Probability of the occurrence of ‘4’

Number of favorable outcomes
P(A) =     Total number of possible outcome

= 75/450
= 1/6

(ii) Number of occurrence of a number less than 4 = 73 + 70 + 74 = 217

Probability of the occurrence of ‘a number < 4’

Number of favorable outcomes
P(B) =     Total number of possible outcome

= 217/450

(iii) Number of occurrence of a number greater than 4 = 80 + 78 = 158

Probability of the occurrence of ‘a number > 4’

Number of favorable outcomes
P(C) =     Total number of possible outcome

= 158/450
= 79/225

(iv) Number of occurrence of a prime number i.e. 2, 3, 5 = 70 + 74 + 80 = 224

Probability of the occurrence of ‘a prime number’

Number of favorable outcomes
P(D) =     Total number of possible outcome

= 224/450
= 112/225

(v) Number of occurrence of a number less than 7 i.e. 1, 2, 3, 4, 5 and 6 = 73 + 70 + 74 + 75 + 80 + 78 = 450

Probability of the occurrence of ‘a number < 7’

Number of favorable outcomes
P(E) =     Total number of possible outcome

= 450/450
= 1

(vi) Number of occurrence of a number greater than 6 = 0,

Because when a die is thrown all the 6 outcomes are 1, 2, 3, 4, 5 and 6

so, there is no number greater than 6.

Probability of the occurrence of ‘a number > 6’

Number of favorable outcomes
P(F) =     Total number of possible outcome

= 0/450
= 0

Probability

Probability

Random Experiments

Experimental Probability

Empirical Probability

Coin Toss Probability

Probability of Tossing Two Coins

Probability of Tossing Three Coins

Complimentary Events

Mutually Exclusive Events

Mutually Non-Exclusive Events

Conditional Probability

Theoretical Probability

Odds and Probability

Playing Cards Probability

Probability and Playing Cards

Solved Probability Problems

Probability for Rolling Three Dice