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
(i) a head
(ii) a tail.

Solution:

Number of times coin is tossed = 450
Number of heads = 250
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

Theoretical Probability

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:

Theoretical Probability Problems

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

9 Grade Math

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