# 5.2 Probability

## Detailed Theory: Probability

### **1. Basic Concepts and Terminology**

#### **1.1 What is Probability?**

Probability is a branch of mathematics that deals with calculating the likelihood of events occurring. It quantifies uncertainty.

**Key Idea:** Probability is a number between 0 and 1 that represents how likely an event is to occur.

* **0** means the event is impossible
* **1** means the event is certain
* **0.5** means the event is equally likely to occur or not occur

#### **1.2 Important Terms**

**a) Experiment**

An action or process that leads to one of several possible outcomes.

**Examples:**

* Tossing a coin
* Rolling a die
* Drawing a card from a deck

**b) Sample Space (S)**

The set of all possible outcomes of an experiment.

**Examples:**

1. Tossing a coin: $$S = {H, T}$$ (H = Head, T = Tail)
2. Rolling a die: $$S = {1, 2, 3, 4, 5, 6}$$
3. Tossing two coins: $$S = {HH, HT, TH, TT}$$

**c) Event (E)**

A subset of the sample space (one or more outcomes).

**Examples for die roll:**

* Event A: Getting an even number = $${2, 4, 6}$$
* Event B: Getting a number > 4 = $${5, 6}$$

**d) Types of Events**

1. **Simple Event:** Single outcome (e.g., getting a 3 on die)
2. **Compound Event:** Multiple outcomes (e.g., getting even number)
3. **Impossible Event:** Event that cannot occur (probability = 0)
4. **Sure/Certain Event:** Event that always occurs (probability = 1)
5. **Complementary Event (E' or E⁰):** All outcomes NOT in E

***

### **2. Different Approaches to Probability**

#### **2.1 Classical (Theoretical) Probability**

Used when all outcomes are equally likely.

**Formula:**

$$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{n(E)}{n(S)}$$

**Example:** Probability of getting an even number when rolling a die:

$$S = {1, 2, 3, 4, 5, 6} \Rightarrow n(S) = 6$$

$$E = {2, 4, 6} \Rightarrow n(E) = 3$$

$$P(E) = \frac{3}{6} = \frac{1}{2} = 0.5$$

#### **2.2 Empirical (Experimental) Probability**

Based on actual experiments or observations.

**Formula:**

$$P(E) = \frac{\text{Number of times event occurred}}{\text{Total number of trials}}$$

**Example:** In 100 coin tosses, heads appeared 47 times.

Empirical probability of heads = $$\frac{47}{100} = 0.47$$

#### **2.3 Axiomatic Probability**

Based on three axioms (rules):

**Axiom 1:** $$0 \leq P(E) \leq 1$$ for any event E

**Axiom 2:** $$P(S) = 1$$ (probability of sample space is 1)

**Axiom 3:** For mutually exclusive events A and B: $$P(A \cup B) = P(A) + P(B)$$

***

### **3. Basic Probability Rules**

#### **3.1 Complement Rule**

The probability that an event does NOT occur.

$$P(E') = 1 - P(E)$$

**Example:** Probability of NOT getting a 6 on a die roll:

$$P(\text{not 6}) = 1 - P(\text{getting 6}) = 1 - \frac{1}{6} = \frac{5}{6}$$

#### **3.2 Addition Rule**

For any two events A and B:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

**Special Case: Mutually Exclusive Events**

Events that cannot occur together (no common outcomes).

For mutually exclusive events: $$P(A \cap B) = 0$$

So: $$P(A \cup B) = P(A) + P(B)$$

**Example:** Probability of getting 2 OR 5 on a die roll:

These are mutually exclusive: $$P(2 \cup 5) = P(2) + P(5) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$$

#### **3.3 Multiplication Rule**

For independent events (occurrence of one doesn't affect the other):

$$P(A \cap B) = P(A) \times P(B)$$

**Example:** Probability of getting heads on two consecutive coin tosses:

$$P(H \text{ and } H) = P(H) \times P(H) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$$

***

### **4. Conditional Probability**

#### **4.1 Definition**

Probability of event A occurring given that event B has already occurred.

**Notation:** $$P(A|B)$$ (read as "probability of A given B")

#### **4.2 Formula**

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$, provided $$P(B) \neq 0$$

This can be rearranged as: $$P(A \cap B) = P(B) \times P(A|B)$$

#### **4.3 Understanding with Example**

In a class of 30 students: 18 girls, 12 boys. 6 girls and 4 boys wear glasses.

Let: A = student wears glasses, B = student is a girl

Find probability that a student wears glasses given they're a girl:

$$P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{6/30}{18/30} = \frac{6}{18} = \frac{1}{3}$$

Direct method: Among 18 girls, 6 wear glasses, so probability = $$\frac{6}{18} = \frac{1}{3}$$

#### **4.4 Independent Events using Conditional Probability**

Events A and B are independent if:

$$P(A|B) = P(A)$$ or $$P(B|A) = P(B)$$

This means knowing B occurred doesn't change probability of A.

**Test for independence:** A and B are independent if and only if: $$P(A \cap B) = P(A) \times P(B)$$

***

### **5. Bayes' Theorem**

#### **5.1 Statement**

For events A and B with $$P(B) \neq 0$$:

$$P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}$$

#### **5.2 Extended Form**

If $$B\_1, B\_2, \ldots, B\_n$$ form a partition of sample space, then:

$$P(B\_i|A) = \frac{P(A|B\_i) \times P(B\_i)}{\sum\_{j=1}^{n} P(A|B\_j) \times P(B\_j)}$$

#### **5.3 Application Example**

**Medical Test Problem:**

* Disease affects 1% of population (P(D) = 0.01)
* Test is 95% accurate: P(positive|D) = 0.95, P(negative|no D) = 0.95
* Find P(D|positive)

**Solution using Bayes':**

Let D = has disease, T+ = test positive

We want: $$P(D|T+) = \frac{P(T+|D) \times P(D)}{P(T+)}$$

$$P(T+|D) = 0.95$$, $$P(D) = 0.01$$

$$P(T+) = P(T+|D)P(D) + P(T+|no D)P(no D)$$ $$= 0.95 \times 0.01 + 0.05 \times 0.99 = 0.0095 + 0.0495 = 0.059$$

So: $$P(D|T+) = \frac{0.95 \times 0.01}{0.059} = \frac{0.0095}{0.059} \approx 0.161$$

Only 16.1% chance of having disease given positive test!

***

### **6. Random Variables**

#### **6.1 Definition**

A variable whose values are determined by outcomes of a random experiment.

**Notation:** Capital letters (X, Y, Z) for random variables, lowercase (x, y, z) for specific values.

#### **6.2 Types of Random Variables**

**a) Discrete Random Variable**

Takes countable number of distinct values.

**Examples:**

* Number of heads in 3 coin tosses (0, 1, 2, 3)
* Number of students absent in a class

**b) Continuous Random Variable**

Takes infinitely many values in an interval (measurable).

**Examples:**

* Height of students
* Time taken to complete a task
* Temperature

#### **6.3 Probability Distribution**

For a discrete random variable X, lists all possible values and their probabilities.

**Requirements:**

1. $$0 \leq P(X = x\_i) \leq 1$$ for all i
2. $$\sum P(X = x\_i) = 1$$

**Example:** Probability distribution of number of heads in 2 coin tosses:

| X (number of heads) | P(X)  |
| ------------------- | ----- |
| 0                   | 1/4   |
| 1                   | 1/2   |
| 2                   | 1/4   |
| **Total**           | **1** |

***

### **7. Expectation (Mean) and Variance**

#### **7.1 Expected Value (Mean)**

Average value we expect if experiment is repeated many times.

For discrete random variable X:

$$E(X) = \mu = \sum x\_i \cdot P(X = x\_i)$$

**Example:** Expected number of heads in 2 coin tosses:

$$E(X) = 0 \times \frac{1}{4} + 1 \times \frac{1}{2} + 2 \times \frac{1}{4} = 0 + 0.5 + 0.5 = 1$$

We expect 1 head on average.

#### **7.2 Properties of Expectation**

1. $$E(c) = c$$ where c is constant
2. $$E(cX) = cE(X)$$
3. $$E(X + Y) = E(X) + E(Y)$$
4. $$E(X - Y) = E(X) - E(Y)$$
5. For independent X, Y: $$E(XY) = E(X)E(Y)$$

#### **7.3 Variance**

Measures how spread out the values are from the mean.

**Formula:** $$\text{Var}(X) = E\[(X - \mu)^2] = E(X^2) - \[E(X)]^2$$

**Calculation formula:** $$\text{Var}(X) = \sum x\_i^2 P(X = x\_i) - \[\sum x\_i P(X = x\_i)]^2$$

#### **7.4 Standard Deviation**

Square root of variance: $$\sigma = \sqrt{\text{Var}(X)}$$

#### **7.5 Properties of Variance**

1. $$\text{Var}(c) = 0$$ where c is constant
2. $$\text{Var}(cX) = c^2 \text{Var}(X)$$
3. $$\text{Var}(X + c) = \text{Var}(X)$$
4. For independent X, Y: $$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$$

***

### **8. Important Probability Distributions**

#### **8.1 Binomial Distribution**

**a) Conditions for Binomial Experiment**

1. Fixed number of trials (n)
2. Each trial has exactly two outcomes: success or failure
3. Probability of success (p) is constant for each trial
4. Trials are independent

**b) Probability Mass Function**

For random variable X = number of successes in n trials:

$$P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}, \quad r = 0, 1, 2, \ldots, n$$

where $$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$ (binomial coefficient)

**c) Mean and Variance**

Mean: $$\mu = np$$

Variance: $$\sigma^2 = np(1-p)$$

Standard Deviation: $$\sigma = \sqrt{np(1-p)}$$

**d) Example**

Probability of getting exactly 3 heads in 5 coin tosses:

$$n=5$$, $$p=0.5$$, $$r=3$$

$$P(X=3) = \binom{5}{3} (0.5)^3 (0.5)^2 = 10 \times 0.125 \times 0.25 = 10 \times 0.03125 = 0.3125$$

#### **8.2 Poisson Distribution**

**a) When to Use**

Models number of events occurring in a fixed interval of time/space, when events occur independently at constant average rate.

**Examples:**

* Number of calls at a call center per hour
* Number of accidents at an intersection per day

**b) Probability Mass Function**

$$P(X = r) = \frac{e^{-\lambda} \lambda^r}{r!}, \quad r = 0, 1, 2, \ldots$$

where $$\lambda$$ = average number of events in the interval

**c) Mean and Variance**

Mean = Variance = $$\lambda$$

#### **8.3 Normal Distribution (Gaussian Distribution)**

**a) Characteristics**

* Bell-shaped curve
* Symmetric about mean
* Mean = median = mode
* Total area under curve = 1

**b) Probability Density Function**

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}, \quad -\infty < x < \infty$$

where $$\mu$$ = mean, $$\sigma$$ = standard deviation

**c) Standard Normal Distribution**

Special case with $$\mu = 0$$, $$\sigma = 1$$

**Z-score:** $$z = \frac{x - \mu}{\sigma}$$

Converts any normal distribution to standard normal.

**d) Empirical Rule (68-95-99.7 Rule)**

For normal distribution:

* About 68% of data falls within 1σ of mean
* About 95% of data falls within 2σ of mean
* About 99.7% of data falls within 3σ of mean

***

### **9. Permutations and Combinations in Probability**

#### **9.1 Fundamental Counting Principle**

If event A can occur in m ways, and event B can occur in n ways, then:

* A AND B can occur in m × n ways
* A OR B can occur in m + n ways

#### **9.2 Permutations**

Arrangements where order matters.

**Number of permutations of n distinct objects taken r at a time:**

$$P(n, r) = \frac{n!}{(n-r)!}$$

**Special case:** All n objects: $$P(n, n) = n!$$

#### **9.3 Combinations**

Selections where order doesn't matter.

**Number of combinations of n distinct objects taken r at a time:**

$$C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$$

#### **9.4 Application in Probability**

**Example:** Probability of getting a specific poker hand.

***

### **10. Solved Examples**

#### **Example 1:** Basic Probability

A bag contains 5 red, 3 blue, and 2 green balls. One ball is drawn at random. Find: a) P(red) b) P(not blue) c) P(red or green)

**Solution:** Total balls = 5 + 3 + 2 = 10

a) $$P(\text{red}) = \frac{5}{10} = 0.5$$

b) $$P(\text{not blue}) = 1 - P(\text{blue}) = 1 - \frac{3}{10} = \frac{7}{10} = 0.7$$

c) $$P(\text{red or green}) = P(\text{red}) + P(\text{green}) = \frac{5}{10} + \frac{2}{10} = \frac{7}{10} = 0.7$$

#### **Example 2:** Conditional Probability

In a class, 60% are girls. 40% of girls and 30% of boys wear glasses. If a student wears glasses, what's the probability they're a girl?

**Solution:** Let: G = girl, B = boy, W = wears glasses

Given: $$P(G) = 0.6$$, $$P(B) = 0.4$$, $$P(W|G) = 0.4$$, $$P(W|B) = 0.3$$

We want: $$P(G|W)$$

By Bayes': $$P(G|W) = \frac{P(W|G)P(G)}{P(W)}$$

$$P(W) = P(W|G)P(G) + P(W|B)P(B)$$ $$= 0.4 \times 0.6 + 0.3 \times 0.4 = 0.24 + 0.12 = 0.36$$

So: $$P(G|W) = \frac{0.4 \times 0.6}{0.36} = \frac{0.24}{0.36} = \frac{2}{3} \approx 0.667$$

#### **Example 3:** Binomial Distribution

A test has 10 multiple-choice questions, each with 4 options. If a student guesses randomly, find probability of: a) Exactly 5 correct b) At least 8 correct

**Solution:** $$n=10$$, $$p = \frac{1}{4} = 0.25$$ (probability of correct guess)

a) $$P(X=5) = \binom{10}{5} (0.25)^5 (0.75)^5$$

$$= 252 \times 0.0009766 \times 0.2373 = 252 \times 0.0002318 = 0.0584$$

b) $$P(X \geq 8) = P(X=8) + P(X=9) + P(X=10)$$

$$P(X=8) = \binom{10}{8} (0.25)^8 (0.75)^2 = 45 \times 0.00001526 \times 0.5625 = 0.000386$$ $$P(X=9) = \binom{10}{9} (0.25)^9 (0.75)^1 = 10 \times 0.000003815 \times 0.75 = 0.0000286$$ $$P(X=10) = \binom{10}{10} (0.25)^{10} (0.75)^0 = 1 \times 0.000000954 = 0.000000954$$

Sum = 0.000386 + 0.0000286 + 0.000000954 = 0.0004155

#### **Example 4:** Expected Value

A game: Roll a die. If you get 6, win $10. If you get 4 or 5, win $5. Otherwise, lose $3. What's expected value?

**Solution:** Let X = winnings

| Outcome | X (winnings) | P(X) |
| ------- | ------------ | ---- |
| 6       | $10          | 1/6  |
| 4 or 5  | $5           | 2/6  |
| 1,2,3   | -$3          | 3/6  |

$$E(X) = 10 \times \frac{1}{6} + 5 \times \frac{2}{6} + (-3) \times \frac{3}{6}$$ $$= \frac{10}{6} + \frac{10}{6} - \frac{9}{6} = \frac{11}{6} \approx $1.83$$

Positive expected value means game is favorable on average.

***

### **11. Important Formulas Summary**

#### **11.1 Basic Probability**

* $$P(E) = \frac{n(E)}{n(S)}$$
* $$P(E') = 1 - P(E)$$
* $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
* Mutually exclusive: $$P(A \cap B) = 0$$

#### **11.2 Conditional Probability**

* $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
* Independent events: $$P(A \cap B) = P(A)P(B)$$

#### **11.3 Bayes' Theorem**

* $$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$

#### **11.4 Random Variables**

* Expected value: $$E(X) = \sum x\_i P(X = x\_i)$$
* Variance: $$\text{Var}(X) = E(X^2) - \[E(X)]^2$$

#### **11.5 Binomial Distribution**

* $$P(X=r) = \binom{n}{r} p^r (1-p)^{n-r}$$
* Mean: $$np$$
* Variance: $$np(1-p)$$

#### **11.6 Poisson Distribution**

* $$P(X=r) = \frac{e^{-\lambda} \lambda^r}{r!}$$
* Mean = Variance = $$\lambda$$

#### **11.7 Counting**

* Permutations: $$P(n,r) = \frac{n!}{(n-r)!}$$
* Combinations: $$C(n,r) = \frac{n!}{r!(n-r)!}$$

***

### **12. Exam Tips and Common Mistakes**

#### **12.1 Common Mistakes to Avoid**

1. **Confusing "and" vs "or":**
   * "A and B" means both occur (intersection)
   * "A or B" means at least one occurs (union)
2. **Misapplying addition rule:** Remember to subtract intersection unless events are mutually exclusive
3. **Confusing independent vs mutually exclusive:**
   * Independent: $$P(A \cap B) = P(A)P(B)$$
   * Mutually exclusive: $$P(A \cap B) = 0$$
4. **Forgetting to check conditions** for binomial/Poisson distributions
5. **Probability > 1 or < 0:** Impossible! Check calculations if this happens

#### **12.2 Problem-Solving Strategy**

1. **Define events clearly:** Write what each event represents
2. **Identify what's asked:** "Given that", "and", "or", etc.
3. **Choose correct formula:** Based on conditions
4. **Check independence/mutual exclusivity**
5. **Draw Venn diagrams or tree diagrams** for visualization

#### **12.3 Quick Checks**

1. **Total probability always sums to 1**
2. **Conditional probability:** $$0 \leq P(A|B) \leq 1$$
3. **Complement:** $$P(A) + P(A') = 1$$
4. **Expected value interpretation:** Long-run average
5. **Variance:** Always non-negative

***

### **13. Real-World Applications**

#### **13.1 Everyday Life**

1. **Weather forecasting:** Probability of rain
2. **Games of chance:** Cards, dice, lotteries
3. **Insurance:** Calculating premiums based on risk

#### **13.2 Science and Engineering**

1. **Quality control:** Probability of defective items
2. **Reliability engineering:** Probability of system failure
3. **Medical testing:** Sensitivity and specificity

#### **13.3 Finance**

1. **Stock market:** Probability of price movements
2. **Risk assessment:** Probability of loan default
3. **Portfolio management:** Expected returns

This comprehensive theory covers all aspects of probability with detailed explanations and examples, making it easy to understand while being thorough enough for exam preparation.


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