Probability – Class 10 Notes
1. Introduction to Probability
- Probability is a branch of mathematics that deals with the likelihood of occurrence of an event.
- It helps predict how likely it is for a particular event to happen when there is uncertainty.
2. Key Terms in Probability
- Experiment: An action or process that leads to one or more outcomes.
- Example: Rolling a die or flipping a coin.
- Random Experiment: An experiment where the outcome is not known in advance.
- Example: Picking a card from a shuffled deck.
- Sample Space (S): The set of all possible outcomes of an experiment.
- Example: When rolling a die, ( S = {1, 2, 3, 4, 5, 6} ).
- Event (E): A subset of the sample space.
- Example: Getting an even number on a die, ( E = {2, 4, 6} ).
3. Calculating Probability
The probability of an event happening is given by:
[
P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
]
Key Points:
- ( P(E) ) will always lie between 0 and 1:
( 0 \leq P(E) \leq 1 ) - If ( P(E) = 0 ): The event is impossible.
- If ( P(E) = 1 ): The event is certain to occur.
4. Types of Events
- Sure Event: An event that will definitely happen.
Example: Getting a number less than 7 on a die roll. - Impossible Event: An event that cannot happen.
Example: Getting a 7 on a standard die. - Complementary Event: The event that the original event does not occur.
Example: If ( E ) is the event of getting an even number, then ( E’ ) (complementary event) is getting an odd number.
5. Examples
- Example 1: What is the probability of getting a 3 when a die is rolled?
Solution:
- Total outcomes: ( S = {1, 2, 3, 4, 5, 6} )
- Favorable outcome: ( {3} )
- ( P(E) = \frac{1}{6} )
- Example 2: What is the probability of drawing a red card from a deck of 52 cards?
Solution:
- Total cards: 52
- Red cards: 26
- ( P(E) = \frac{26}{52} = \frac{1}{2} )
6. Tips for Solving Problems
- List the sample space clearly for each problem.
- Identify the favorable outcomes that match the event.
- Apply the formula for probability carefully.
- Check if events are complementary to use ( P(E) + P(E’) = 1 ).
7. Important Formula
- ( P(E) + P(E’) = 1 )
With practice, you will get comfortable calculating probabilities for various experiments. Keep solving examples to sharpen your understanding!
Additional Example Problems on Probability
Additional Example Problems on Probability
Example 3: Flipping a Coin
Problem:
What is the probability of getting a tail when a fair coin is tossed?
Solution:
- Sample Space, S = {Head (H), Tail (T)}
- Favorable Outcome = {T}
- Total Number of Outcomes = 2
- ( P(\text{Tail}) = \frac{1}{2} )
Example 4: Rolling a Die
Problem:
What is the probability of getting a number greater than 4 on a standard 6-sided die?
Solution:
- Sample Space, S = {1, 2, 3, 4, 5, 6}
- Favorable Outcomes = {5, 6}
- Number of Favorable Outcomes = 2
- Total Number of Outcomes = 6
[
P(\text{number greater than 4}) = \frac{2}{6} = \frac{1}{3}
]
Example 5: Drawing a Card from a Deck
Problem:
What is the probability of drawing a King from a standard deck of 52 cards?
Solution:
- Total Cards = 52
- Number of Kings = 4
- ( P(\text{King}) = \frac{4}{52} = \frac{1}{13} )
Example 6: Probability of an Odd Number on a Die
Problem:
What is the probability of getting an odd number when a die is rolled?
Solution:
- Sample Space, S = {1, 2, 3, 4, 5, 6}
- Favorable Outcomes = {1, 3, 5}
- Number of Favorable Outcomes = 3
- Total Number of Outcomes = 6
[
P(\text{Odd number}) = \frac{3}{6} = \frac{1}{2}
]
Example 7: Two Coin Tosses
Problem:
What is the probability of getting exactly one head when two coins are tossed?
Solution:
- Sample Space, S = {HH, HT, TH, TT}
- Favorable Outcomes = {HT, TH}
- Number of Favorable Outcomes = 2
- Total Number of Outcomes = 4
[
P(\text{Exactly one head}) = \frac{2}{4} = \frac{1}{2}
]
Example 8: Drawing a Red Ball from a Bag
Problem:
A bag contains 5 red balls, 3 blue balls, and 2 green balls. What is the probability of drawing a red ball?
Solution:
- Total Balls = 5 + 3 + 2 = 10
- Favorable Outcomes (Red Balls) = 5
[
P(\text{Red ball}) = \frac{5}{10} = \frac{1}{2}
]
Example 9: Probability of Not Drawing an Ace
Problem:
What is the probability of not drawing an Ace from a deck of 52 cards?
Solution:
- Total Cards = 52
- Number of Aces = 4
- Number of Non-Ace Cards = 52 – 4 = 48
[
P(\text{Not an Ace}) = \frac{48}{52} = \frac{12}{13}
]
Example 10: Tossing Three Coins
Problem:
What is the probability of getting at least one head when three coins are tossed?
Solution:
- Sample Space, S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
- Number of Total Outcomes = 8
- Favorable Outcomes (At least one head) = 7 (all except TTT)
[
P(\text{At least one head}) = \frac{7}{8}
]
Example 11: Rolling Two Dice
Problem:
Two dice are rolled. What is the probability of getting a sum of 7?
Solution:
- Sample Space, S: There are (6 \times 6 = 36) total outcomes (since each die has 6 faces).
- Favorable Outcomes (Sum = 7):
- (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
- Number of Favorable Outcomes = 6
[
P(\text{Sum of 7}) = \frac{6}{36} = \frac{1}{6}
]
Example 12: Selecting a Student Randomly
Problem:
A class has 30 students: 12 are boys, and 18 are girls. What is the probability of selecting a girl randomly?
Solution:
- Total Students = 30
- Favorable Outcomes (Girls) = 18
[
P(\text{Girl}) = \frac{18}{30} = \frac{3}{5}
]
These examples cover a variety of scenarios to help you understand and practice probability effectively! Keep practicing and work through similar problems to strengthen your concepts.