Elementary Theorems on Probability

Introduction

The probability of any event is the possible chances of occurrence of any outcome. For example, you know that the only two possible chances are a head and a tail when you toss a coin. This event has two possible outcomes, each with a 50% probability. Mathematically, the probability is the ratio of the number of possible outcomes to the number of total outcomes. The probability of any outcome lies between 0 to 1. This is so because the favorable number of outcomes can never cross the total number of outcomes in an event. If the probability of a certain event is 0, then that event is impossible. On the other hand, if the probability of a certain event is 1, it will be a sure event. Let’s know about the elementary theorems on probability.

Probability Theorems

Different elementary theorems on probability help us in understanding and easy solving probability questions:

Theorem 1: The sum of probability of happening and not happening of any given event is always unity, i.e., equals 1.

P(E) + P(E’) = 1

Theorem 2: The probability of an impossible event is always equal to 0.

P(E) = 0

Theorem 3: The sure events always have 1 as a probability.

P(E) = 1

Theorem 4: The probability of any event is always between 0 to 1.

0 < P(E) < 1

Theorem 5: For any two events, say M and N, the derivation of the formula of probability can be made by applying the formula of the union of two sets.

Therefore, the formula of event M and N is as follows:

P(M∪N) = P(M) + P(N) − P(M∩N)

Two mutually exclusive events = P ( L U M) = P(L) + P(M)

Bayes’ Theorem: Bayes’ theorem or conditional probability theorem defines or states the probability of any event based on conditions of other events.

The formula for Bayes’ theorem based on conditional probability is

P (M|N) = Occurrence of event M on the basis or condition of event N.

Law of Total Probability: If any experiment has multiple numbers of events, then the sum of all these events will always be equal to 1.

The formula of the law of total probability is

P(A1) + P(A2) + P(A3) + P(A4) + P(A5) + …. + P(An) = 1

Solved Questions on Probability

1. What is the probability of getting a prime number while rolling a dice in a game of Ludo?

Solution:

The probability of getting a prime number = P(A)

Total number of outcomes = 6

Number of favourable outcomes or desired outcomes= 3 [2, 3, 5]

Probability = P(A)= = 3/6

= ½

Answer: The probability of getting a prime number while rolling dice is ½, i.e., 50%.

2. There is a box of candies with 4 chocolate candies, 3 strawberry candies, and 3 raspberry candies. What is the probability of getting chocolate candy?

Solution:

The probability of getting a chocolate candy = P(A)

Total number of outcomes = 10

Number of favorable outcomes or desired outcomes = 4

Probability = P(A)

= 4/10

= ⅖

Answer: The probability of getting a chocolate candy is ⅖.

3. What is the probability of getting a heart card from a standard deck of cards?

Solution:

The probability of getting a heart card = P(A)

Total number of outcomes = 52 (total number of cards)

Number of favorable outcomes or desired outcomes = 13 (total number of heart cards in a deck)

Probability = P(A)

= 13/52

= ¼

Answer: The probability of getting a heart will be ¼, i.e., 25%.

Conclusion

The probability of any event is the possible chances of occurrence of any outcome. Mathematically, the probability is the ratio of the number of possible outcomes to the number of total outcomes. The probability of any outcome lies between 0 to 1 because the favorable number of outcomes can never cross the total number of outcomes in an event. If the probability of a particular event is 0, then that event is impossible. On the other hand, if the probability of a specific event is 1, it will be a sure event. Different elementary theorems on probability help us understand and quickly solve probability questions. Bayes’ theorem or conditional probability theorem defines or states the probability of any event based on conditions of other events.