Probability
Complete Chapter 13 notes covering conditional probability, multiplication theorem, independent events, Bayes' theorem, random variables, probability distributions, mean, variance, solved examples and examination practice.
1. Introduction to Probability
Probability is the mathematical measure of the likelihood of occurrence of an event. In Class XII, probability is extended to conditional probability, independent events, Bayes' theorem and probability distributions.
| Concept | Meaning |
|---|---|
| Experiment | A process producing well-defined outcomes. |
| Sample Space | Set of all possible outcomes. |
| Event | A subset of the sample space. |
| Sure Event | An event that always occurs. |
| Impossible Event | An event that cannot occur. |
2. Conditional Probability
Conditional probability is the probability of an event occurring when it is known that another event has already occurred.
P(B | A) = P(A ∩ B) / P(A), P(A) ≠ 0
Important Result
= P(A)P(B | A)
The condition after the vertical bar is the event that is assumed to have occurred.
3. Multiplication Theorem of Probability
The multiplication theorem is used to find the probability of the simultaneous occurrence of two events.
P(A ∩ B) = P(B)P(A | B)
For Three Events
= P(A)P(B | A)P(C | A ∩ B)
4. Independent Events
Two events A and B are independent if the occurrence of one event does not affect the probability of the other event.
For independent events:
P(B | A) = P(B)
Three Independent Events
Mutually exclusive events with non-zero probabilities cannot be independent.
5. Theorem of Total Probability
Suppose E₁, E₂, ..., Eₙ are mutually exclusive and exhaustive events, and A is another event.
- E₁, E₂, ..., Eₙ are mutually exclusive.
- E₁ ∪ E₂ ∪ ... ∪ Eₙ = S.
- P(Eᵢ) > 0.
6. Bayes' Theorem
Bayes' theorem is used to calculate the probability of a particular cause when the effect is known.
Standard Bayes' Theorem Form
- Medical testing problems.
- Factory or machine production problems.
- Selection from different boxes or bags.
- Source identification problems.
7. Random Variable
A random variable is a real-valued function defined on the sample space of a random experiment.
When a coin is tossed three times, let X denote the number of heads obtained.
Types of Random Variables
Takes a finite or countably infinite number of values.
Can take values over an interval.
8. Probability Distribution
If X is a discrete random variable taking values x₁, x₂, ..., xₙ with probabilities p₁, p₂, ..., pₙ, then the probability distribution is represented by a table.
| X | x₁ | x₂ | ... | xₙ |
|---|---|---|---|---|
| P(X) | p₁ | p₂ | ... | pₙ |
Σpᵢ = 1
9. Mean, Variance and Standard Deviation
Mean / Expected Value
Variance
Expected Value of X²
Standard Deviation
10. Binomial Distribution
In a binomial experiment, there are a fixed number of independent trials, each trial has two possible outcomes, and the probability of success remains constant.
Conditions of Binomial Distribution
- There are a fixed number n of trials.
- Each trial has two possible outcomes: success or failure.
- The trials are independent.
- Probability of success remains constant.
Probability Mass Function
Mean
Variance
Standard Deviation
11. Solved Examples
Example 1: Conditional Probability
If P(A) = 1/2, P(B) = 1/3 and P(A ∩ B) = 1/6 , find P(A | B) .
= (1/6) / (1/3)
= 1/2
Answer: P(A | B) = 1/2
Example 2: Independent Events
If A and B are independent events with P(A) = 0.4 and P(B) = 0.5 , find P(A ∩ B) .
= 0.4 × 0.5
= 0.2
Answer: 0.2
Example 3: Probability Distribution
A random variable X has the following probability distribution:
| X | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X) | 0.1 | 0.2 | 0.3 | 0.4 |
Find E(X).
= 0(0.1) + 1(0.2) + 2(0.3) + 3(0.4)
= 0 + 0.2 + 0.6 + 1.2
= 2
Answer: E(X) = 2
Example 4: Binomial Probability
A fair coin is tossed 4 times. Find the probability of getting exactly 2 heads.
r = 2
p = 1/2
q = 1/2
P(X = 2) = ⁴C₂(1/2)²(1/2)²
= 6/16
= 3/8
Answer: 3/8
Example 5: Bayes' Theorem
A product is manufactured by two machines. The probability that a randomly selected product comes from Machine A is 0.6 and from Machine B is 0.4. Their defect probabilities are 0.02 and 0.05 respectively. If a selected product is defective, Bayes' theorem can be used to determine the probability that it was produced by Machine B.
Important Probability Formulas
| Concept | Formula |
|---|---|
| Complement | P(A') = 1 − P(A) |
| Conditional Probability | P(A|B) = P(A∩B)/P(B) |
| Multiplication Theorem | P(A∩B) = P(A)P(B|A) |
| Independent Events | P(A∩B) = P(A)P(B) |
| Total Probability | P(A) = ΣP(Eᵢ)P(A|Eᵢ) |
| Bayes' Theorem | P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ)/P(A) |
| Mean of X | E(X) = Σxᵢpᵢ |
| Variance | Var(X) = E(X²) − [E(X)]² |
| Binomial Probability | P(X=r) = ⁿCᵣpʳqⁿ⁻ʳ |
| Binomial Mean | np |
| Binomial Variance | npq |
12. Practice Questions
- Define conditional probability and derive its formula.
- If P(A) and P(B) are given, find P(A ∩ B) using the multiplication theorem.
- Determine whether two given events are independent.
- Three coins are tossed simultaneously. Find the probability of obtaining exactly two heads.
- A box contains red and blue balls. Two balls are drawn successively. Find the probability using conditional probability.
- Apply the theorem of total probability to a problem involving two or more machines.
- Solve a Bayes' theorem problem involving defective products.
- Construct the probability distribution of the number of heads obtained when a coin is tossed three times.
- Find the mean and variance of a given probability distribution.
- A binomial random variable has n = 10 and p = 0.4. Find its mean and variance.
- Find the probability of obtaining exactly three successes in a specified number of independent trials.
- Solve a mixed problem involving conditional probability and Bayes' theorem.
13. Higher-Order Thinking (HOT) Questions
Two events have equal probabilities. Determine conditions under which they can be both independent and mutually exclusive.
A product is selected from one of several factories. Given that the product is defective, determine the probability that it came from a particular factory using Bayes' theorem.
A random variable has an unknown parameter in its probability distribution. Use the condition that the sum of all probabilities is 1 to determine the parameter.
Find a parameter such that the mean and variance of a probability distribution satisfy a specified relationship.
A binomial distribution has a specified mean and variance. Determine the values of n and p.
Compare two probability distributions and determine which one has greater variability using variance.
14. Quick Revision Formula Sheet
P(A|B) = P(A∩B)/P(B)
P(A∩B) = P(A)P(B|A)
P(A∩B) = P(A)P(B)
P(A) = ΣP(Eᵢ)P(A|Eᵢ)
P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ)/P(A)
E(X) = Σxᵢpᵢ
Var(X) = E(X²) − [E(X)]²
P(X=r) = ⁿCᵣpʳqⁿ⁻ʳ
q = 1 − p
np
npq
15. Examination Tips
- Clearly identify the sample space and events before applying a probability formula.
- For conditional probability, carefully identify the event appearing after the vertical bar.
- Use the multiplication theorem when simultaneous occurrence of events is required.
- Check the independence condition before using P(A ∩ B) = P(A)P(B).
- In Bayes' theorem questions, clearly identify the possible causes and the observed event.
- For a probability distribution, always verify that the sum of all probabilities is 1.
- For a random variable, calculate E(X) and E(X²) carefully before finding variance.
- In binomial distribution problems, identify n, r, p and q before substituting.
- Remember: q = 1 − p .
- Write the final probability in the simplest exact form whenever possible.
- Confusing P(A|B) with P(B|A).
- Applying the multiplication theorem incorrectly.
- Assuming events are independent without checking the condition.
- Using Bayes' theorem without identifying the complete set of possible causes.
- Forgetting that all probabilities in a probability distribution must add up to 1.
- Confusing variance with standard deviation.
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