The chance of occurrence of an event is called probability. The probability values are in the range between 0 and 1. Probability can be studied using two approaches theoretical probability and experimental probability. We will discuss the definition, examples, formula and solved problems of theoretical probability in the following sections.
Theoretical Probability Definition
Theoretical probability defines the theory behind probability. It is not mandatory to conduct an experiment to find the probability of an event with the theoretical probability. Theoretical probability is the ratio of the number of favourable outcomes to the total number of possible outcomes. It is also known as the classical probability or priori probability.
The theoretical probability formula is given as:
Probability of an event = P(E) = \(\frac { Number of favourable outcomes }{ Total number of possible outcomes } \)
Examples:
1. Find the probability when a die is rolled, it rolls a 5.
The number of possible outcomes = 6
Number of favourable outcomes = Number of times a fair die can roll 4 in a single throw = 1
P(E) = \(\frac { Number of favourable outcomes }{ Total number of possible outcomes } \)
P(a fair die rolls 5 in a throw) = \(\frac { 1 }{ 6 } \)
2. A fair coin is tossed 10 times and the outcomes were recorded as Head = 4, tail = 6. Find the probability of the coin showing tail.
Number of times coin tossed = 10
Number of tails = 6
P(E) = \(\frac { Number of favourable outcomes }{ Total number of possible outcomes } \)
P(tail) = \(\frac { 6 }{ 10 } \) = \(\frac { 3 }{ 5 } \)
Also, Check
Difference Between Theoretical Probability & Experimental Probability
| Theoretical Probability | Experimental Probability |
|---|---|
| It is defined as the ratio of the number of favourable outcomes to the number of possible outcomes. | It is defined as the ratio of the number of times an event occurs to the total number of trials. |
| It is also known as classical probability. | It is also called as an empirical probability. |
| Formula is \(\frac { Number of favourable outcomes }{ Total number of possible outcomes } \) | Formula is \(\frac { Number of times event occurs }{ Total number of trials } \) |
| The trials are done only once. | Here, the number of trials are extremely high. |
Questions on Theoretical Probability
Question 1:
Find the probability of getting a composite number in a throw of a die.
Solution:
Let E be the event of getting a composite number.
The total number of possible outcomes = 6 (Since any one of 1, 2, 3, 4, 5, 6 can come).
A number of favourable outcomes for the event E = 2 (Since any one of 4, 6 is a composite number).
So, P(E) = \(\frac { Number of favourable outcomes }{ Total number of possible outcomes } \)
= \(\frac { 2 }{ 6 } \) = \(\frac { 1 }{ 3 } \)
Question 2:
In a society, 1000 families with 2 children were selected and the following data were recorded.
| Number of boys in a family | 0 | 1 | 2 |
| Number of families | 392 | 333 | 275 |
Find the probability of a family, having:
(i) 1 boy
(ii) 2 boys
(iii) no boy.
Solution:
Total number of families = 333 + 392 + 275 = 1000
Number of families having 0 boy = 392
Number of families having 1 boy = 333
Number of families having 2 boys = 275
(i) Probability of having ‘1 boy’
P(X) = \(\frac { Number of favourable outcomes }{ Total number of possible outcomes } \)
= \(\frac { 333 }{ 1000 } \)
(ii) Probability of having ‘2 boys’
P(Y) = \(\frac { Number of favourable outcomes }{ Total number of possible outcomes } \)
= \(\frac { 275 }{ 1000 } \)
(iii) Probability of having ‘0 boys’
P(Z) = \(\frac { Number of favourable outcomes }{ Total number of possible outcomes } \)
= \(\frac { 392 }{ 1000 } \)
FAQ’s on Classical Probability
1. How to calculate theoretical probability?
The formula to calculate the theoretical probability is \(\frac { Number of favourable outcomes }{ Total number of possible outcomes } \). Substitute the values in the formula and find the ratio to get the theoretical probability answer.
2. What are some examples of theoretical probability?
Some of the examples of theoretical probability are rolling a die, flipping a coin, and checking the outcomes. Another example is picking a card from the pack of 52 cards.
3. What does theoretical probability mean?
Theoretical probability calculates the probability of happening, not actually going out and experimenting.