Probability comes into application in the fields of physical sciences, commerce, biological sciences, medical sciences, weather forecasting, etc. As far as the JEE exam is concerned, the probability is an important topic to score more marks with fewer calculations. Probability refers to the occurrence of a random event. It is used to predict how likely the events will happen. In this article, we will see how to find the probability of two events.
When Events are Independent
When the outcome of the first event does not influence the outcome of the second event, those events are called independent events. To determine the probability of two independent events, we have to multiply the probability of the first event by the probability of the second event.
| If A and B are two independent events, then the probability of both happening is given by the formula:
P(A and B) = P(A) × P(B) |
Example
Two dice, one coloured green and one coloured red, are thrown. Find the probability that the score on the red die is 3, and the green die is 1.
Solution:
Number of outcomes when a die is thrown = 6
Probability of score on the red die is 3 = 1/6
Probability of score on the green die is 1 = 1/6
The probability that red shows 3 and green shows 1 = (1/6) × (1/6) = 1/36
Hence, the required probability is 1/36.
Probability of Independent Events – Solved Question
When Events are Dependent
When the outcome of the first event influences the outcome of the second event, those events are called dependent events. The formula to get the probability of dependent events is given below.
| If A and B are two dependent events, then P(A and B) = P(A) × P(B/A)
Here, P(B/A) denotes the probability of B once A has happened. |
Example:
What is the probability of choosing two black cards in a deck of cards (without replacement)?
Solution:
There are 26 red and 26 black cards in a deck of cards.
Probability of choosing a black card at random = 26/52 = 1/2
Probability of choosing a black card again = 25/51
Required probability = (1/2) × (25/52) = 25/104
When Events are Mutually Exclusive
Two events are said to be mutually exclusive when they cannot happen simultaneously. The sum of their individual probabilities gives the probability that one of the mutually exclusive events will occur.
| If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B) |
Example:
A die is thrown. Find the probability of the die showing a number 2 or number 5.
Solution:
Probability of getting 2 = 1/6
Probability of getting 5 = 1/6
Probability of getting 2 or 5 = (1/6) + (1/6) = 2/6 = 1/3
Hence, the required probability is 1/3.
Related Links:
- Bayes Theorem of Probability
- Probability Problems
- Probability JEE Main Previous Year Questions With Solutions
Frequently Asked Questions
What are independent events?
A and B are independent events, if the probability of occurrence of an event A is not affected by the occurrence of another event B.
How to find the probability of two independent Events?
To determine the probability of two independent events, we have to multiply the probability of the first event by the probability of the second event.
What are dependent events?
When the outcome of the first event influences the outcome of the second event, those events are called dependent events.
What is the probability of both A and B happening, if A and B are two independent events?
If A and B are two independent events, then the probability of both happening is given by the equation, P(A and B) = P(A) × P(B).
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