Introduction to Probability:
Probability is a branch of mathematics that deals with the likelihood of an event occurring.
It is expressed as a value between 0 and 1, where 0 represents an impossible event, and 1 represents a certain event.
The probability of an event A is denoted by P(A).
Types of Probability:
Theoretical Probability:
Theoretical probability is based on theoretical calculations and assumptions. It is the ratio of the number of favorable outcomes to the total number of possible outcomes.
Formula: P(A) = Number of favorable outcomes / Total number of possible outcomes.
Experimental Probability:
Experimental probability is based on actual experimentation or observation.
It is the ratio of the number of times an event occurs to the total number of trials. Formula: P(A) = Number of times event A occurs / Total number of trials.
Probability of an Event:
The probability of an event A is given by the ratio of the number of favorable outcomes to the total number of outcomes.
Example 1:
Rolling a fair six-sided die, find the probability of getting a 3.
Solution:
Total number of possible outcomes = 6 (since there are 6 sides on the die).
Number of favorable outcomes = 1 (there is only one side with a 3).
Probability of getting a 3 = 1/6.
Example 2:
Drawing a card from a standard deck of 52 cards, find the probability of drawing a heart (assuming the deck is well-shuffled).
Solution:
Total number of possible outcomes = 52 (total cards in the deck).
Number of favorable outcomes = 13 (there are 13 hearts in the deck).
Probability of drawing a heart = 13/52 = 1/4.
Addition Rule of Probability:
The addition rule is used to find the probability of the union of two events A and B (P(A ∪ B)).
For mutually exclusive events, the formula is: P(A ∪ B) = P(A) + P(B).
Example 3:
In a bag, there are 5 red balls and 3 blue balls. What is the probability of drawing either a red or a blue ball?
Solution:
Number of red balls = 5.
Number of blue balls = 3.
Total number of balls = 5 + 3 = 8.
Probability of drawing either a red or a blue ball = (Number of red balls + Number of blue balls) / Total number of balls = 5/8 + 3/8 = 8/8 = 1.
Multiplication Rule of Probability:
The multiplication rule is used to find the probability of the intersection of two events A and B (P(A ∩ B)).
For independent events, the formula is: P(A ∩ B) = P(A) * P(B).
Example 4:
A bag contains 4 red balls and 5 blue balls. If two balls are drawn without replacement, find the probability of getting a red ball followed by a blue ball.
Solution:
Number of red balls = 4.
Number of blue balls = 5.
Total number of balls = 4 + 5 = 9.
Probability of getting a red ball in the first draw = Number of red balls / Total number of balls = 4/9.
Probability of getting a blue ball in the second draw (after removing a red ball) = Number of blue balls / Total number of balls = 5/8.
Since the events are not independent (balls are drawn without replacement), we use the multiplication rule:
P(Red followed by Blue) = P(Red) * P(Blue) = (4/9) * (5/8) = 20/72 = 5/18.
Conditional Probability:
Conditional probability is the probability of an event occurring given that another event has already occurred.
The conditional probability of A given B is denoted by P(A|B).
The formula for conditional probability is: P(A|B) = P(A ∩ B) / P(B).
Example 5:
In a class, 60% of students play football, and 40% play basketball. If 30% of the students play both sports, find the probability that a randomly chosen student plays basketball given that the student plays football.
Solution:
Let A be the event "Playing Football" and B be the event "Playing Basketball."
P(A) = 60% = 0.60
P(B) = 40% = 0.40
P(A ∩ B) = 30% = 0.30
The probability of playing basketball given that the student plays football (P(B|A)) is given by:
P(B|A) = P(A ∩ B) / P(A) = 0.30 / 0.60 = 1/2.
These are some of the fundamental concepts and examples related to probability that are typically covered in Class 10 Mathematics. Make sure to practice solving problems and exercises to reinforce your understanding of probability concepts.
Probability is a branch of mathematics that deals with the likelihood of an event occurring.
It is expressed as a value between 0 and 1, where 0 represents an impossible event, and 1 represents a certain event.
The probability of an event A is denoted by P(A).
Types of Probability:
Theoretical Probability:
Theoretical probability is based on theoretical calculations and assumptions. It is the ratio of the number of favorable outcomes to the total number of possible outcomes.
Formula: P(A) = Number of favorable outcomes / Total number of possible outcomes.
Experimental Probability:
Experimental probability is based on actual experimentation or observation.
It is the ratio of the number of times an event occurs to the total number of trials. Formula: P(A) = Number of times event A occurs / Total number of trials.
Probability of an Event:
The probability of an event A is given by the ratio of the number of favorable outcomes to the total number of outcomes.
Example 1:
Rolling a fair six-sided die, find the probability of getting a 3.
Solution:
Total number of possible outcomes = 6 (since there are 6 sides on the die).
Number of favorable outcomes = 1 (there is only one side with a 3).
Probability of getting a 3 = 1/6.
Example 2:
Drawing a card from a standard deck of 52 cards, find the probability of drawing a heart (assuming the deck is well-shuffled).
Solution:
Total number of possible outcomes = 52 (total cards in the deck).
Number of favorable outcomes = 13 (there are 13 hearts in the deck).
Probability of drawing a heart = 13/52 = 1/4.
Addition Rule of Probability:
The addition rule is used to find the probability of the union of two events A and B (P(A ∪ B)).
For mutually exclusive events, the formula is: P(A ∪ B) = P(A) + P(B).
Example 3:
In a bag, there are 5 red balls and 3 blue balls. What is the probability of drawing either a red or a blue ball?
Solution:
Number of red balls = 5.
Number of blue balls = 3.
Total number of balls = 5 + 3 = 8.
Probability of drawing either a red or a blue ball = (Number of red balls + Number of blue balls) / Total number of balls = 5/8 + 3/8 = 8/8 = 1.
Multiplication Rule of Probability:
The multiplication rule is used to find the probability of the intersection of two events A and B (P(A ∩ B)).
For independent events, the formula is: P(A ∩ B) = P(A) * P(B).
Example 4:
A bag contains 4 red balls and 5 blue balls. If two balls are drawn without replacement, find the probability of getting a red ball followed by a blue ball.
Solution:
Number of red balls = 4.
Number of blue balls = 5.
Total number of balls = 4 + 5 = 9.
Probability of getting a red ball in the first draw = Number of red balls / Total number of balls = 4/9.
Probability of getting a blue ball in the second draw (after removing a red ball) = Number of blue balls / Total number of balls = 5/8.
Since the events are not independent (balls are drawn without replacement), we use the multiplication rule:
P(Red followed by Blue) = P(Red) * P(Blue) = (4/9) * (5/8) = 20/72 = 5/18.
Conditional Probability:
Conditional probability is the probability of an event occurring given that another event has already occurred.
The conditional probability of A given B is denoted by P(A|B).
The formula for conditional probability is: P(A|B) = P(A ∩ B) / P(B).
Example 5:
In a class, 60% of students play football, and 40% play basketball. If 30% of the students play both sports, find the probability that a randomly chosen student plays basketball given that the student plays football.
Solution:
Let A be the event "Playing Football" and B be the event "Playing Basketball."
P(A) = 60% = 0.60
P(B) = 40% = 0.40
P(A ∩ B) = 30% = 0.30
The probability of playing basketball given that the student plays football (P(B|A)) is given by:
P(B|A) = P(A ∩ B) / P(A) = 0.30 / 0.60 = 1/2.
These are some of the fundamental concepts and examples related to probability that are typically covered in Class 10 Mathematics. Make sure to practice solving problems and exercises to reinforce your understanding of probability concepts.