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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
- 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%.
- 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.
- 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%.
