Sample Space and Events

When performing a random experiment, the sample space (S) is the set of all possible outcomes. An event is any subset of the sample space.

Example: When a fair die is rolled once, the sample space is $S = \{1, 2, 3, 4, 5, 6\}$. The event of getting an even number is $E = \{2, 4, 6\}$.

Basic Probability Formula

For equally likely outcomes, the probability of an event $E$ is:

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

Example: A fair die is rolled once. What is the probability of getting an even number?

• Sample space: $S = \{1, 2, 3, 4, 5, 6\}$, so $n(S) = 6$
• Event (even number): $E = \{2, 4, 6\}$, so $n(E) = 3$
• $P(E) = \frac{3}{6} = \frac{1}{2} = 0.5$

Probability Axioms

1. Non-negativity: For any event $E$, $0 \leq P(E) \leq 1$
2. Certainty: $P(S) = 1$ (probability of sample space is 1)
3. Addition: For mutually exclusive events $E_1, E_2, \ldots, E_n$: $P(E_1 \cup E_2 \cup \ldots \cup E_n) = P(E_1) + P(E_2) + \ldots + P(E_n)$

Complement of an Event

The complement of event $E$, denoted $E'$, contains all outcomes in $S$ that are not in $E$. The relationship is:

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

Addition Law for Non-Mutually Exclusive Events

For two events $A$ and $B$ that are not mutually exclusive:

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

Independent Events

Two events $A$ and $B$ are independent if the occurrence of one does not affect the occurrence of the other. For independent events:

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

Example: A bag contains 7 red and 8 blue balls. What is the probability of drawing a red ball, replacing it, and then drawing a blue ball?

• $P(\text{red}) = \frac{7}{15}$
• $P(\text{blue}) = \frac{8}{15}$
• Since the ball is replaced, events are independent: $P(\text{red then blue}) = \frac{7}{15} \times \frac{8}{15} = \frac{56}{225}$

A random variable is a variable whose value is determined by the outcome of a random experiment. There are two types:

• Discrete random variable: Takes countable values (e.g., number of heads in coin tosses)
• Continuous random variable: Takes any value in an interval (e.g., height of students)

Probability Distribution of a Discrete Random Variable

For a discrete random variable $X$ taking values $x_1, x_2, \ldots, x_n$ with probabilities $p_1, p_2, \ldots, p_n$:

$\sum_{i=1}^{n} p_i = 1$

Example: A fair coin is tossed twice. Let $X$ be the number of heads obtained. Find the probability distribution.

• Sample space: $S = \{HH, HT, TH, TT\}$
• $X = 0$ (TT): $P(X = 0) = \frac{1}{4}$
• $X = 1$ (HT, TH): $P(X = 1) = \frac{2}{4} = \frac{1}{2}$
• $X = 2$ (HH): $P(X = 2) = \frac{1}{4}$

Check: $\frac{1}{4} + \frac{1}{2} + \frac{1}{4} = 1$ ✓

The expectation or mean of a discrete random variable $X$, denoted $E[X]$ or $\mu$, is the weighted average of all possible values:

$E[X] = \sum_{i=1}^{n} x_i p_i = x_1p_1 + x_2p_2 + \ldots + x_np_n$

Worked Example:

A random variable $X$ has the following probability distribution:

Find $E[X]$.

Solution:

$E[X] = 1(0.1) + 2(0.3) + 3(0.2) + 4(0.3) + 5(0.1)$ $E[X] = 0.1 + 0.6 + 0.6 + 1.2 + 0.5 = 3.0$

Therefore, $E[X] = 3$.

Properties of Expectation

For any random variable $X$ and constants $a$ and $b$:

• $E[aX] = aE[X]$
• $E[X + b] = E[X] + b$
• $E[aX + b] = aE[X] + b$

The variance measures the spread or dispersion of a random variable around its mean. It is defined as:

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

Where: $E[X^2] = \sum_{i=1}^{n} x_i^2 p_i$

Worked Example:

Using the same probability distribution above, find $\text{Var}(X)$.

Solution:

First, find $E[X^2]$: $E[X^2] = 1^2(0.1) + 2^2(0.3) + 3^2(0.2) + 4^2(0.3) + 5^2(0.1)$ $E[X^2] = 1(0.1) + 4(0.3) + 9(0.2) + 16(0.3) + 25(0.1)$ $E[X^2] = 0.1 + 1.2 + 1.8 + 4.8 + 2.5 = 10.4$

Now, using $E[X] = 3$: $\text{Var}(X) = E[X^2] - (E[X])^2 = 10.4 - 3^2 = 10.4 - 9 = 1.4$

Therefore, $\text{Var}(X) = 1.4$.

Standard Deviation

The standard deviation is the square root of the variance:

$\sigma = \sqrt{\text{Var}(X)} = \sqrt{E[X^2] - (E[X])^2}$

For the example above: $\sigma = \sqrt{1.4} \approx 1.183$

Properties of Variance

For any random variable $X$ and constants $a$ and $b$:

• $\text{Var}(aX) = a^2 \text{Var}(X)$
• $\text{Var}(X + b) = \text{Var}(X)$
• $\text{Var}(aX + b) = a^2 \text{Var}(X)$

For a continuous random variable $X$ with probability density function $f(x)$:

• $f(x) \geq 0$ for all $x$
• $\int_{-\infty}^{\infty} f(x) \, dx = 1$
• $P(a \leq X \leq b) = \int_a^b f(x) \, dx$

Expectation of Continuous Random Variable

$E[X] = \int_{-\infty}^{\infty} x f(x) \, dx$

Variance of Continuous Random Variable

$\text{Var}(X) = \int_{-\infty}^{\infty} x^2 f(x) \, dx - \left( \int_{-\infty}^{\infty} x f(x) \, dx \right)^2$

Worked Example:

A continuous random variable $X$ has p.d.f $f(x) = \frac{x}{8}$ for $0 \leq x \leq 4$. Find the mean and variance.

Solution:

Mean: $E[X] = \int_0^4 x \cdot \frac{x}{8} \, dx = \int_0^4 \frac{x^2}{8} \, dx = \left[\frac{x^3}{24}\right]_0^4 = \frac{64}{24} = \frac{8}{3}$

$E[X^2]$: $E[X^2] = \int_0^4 x^2 \cdot \frac{x}{8} \, dx = \int_0^4 \frac{x^3}{8} \, dx = \left[\frac{x^4}{32}\right]_0^4 = \frac{256}{32} = 8$

Variance: $\text{Var}(X) = E[X^2] - (E[X])^2 = 8 - \left(\frac{8}{3}\right)^2 = 8 - \frac{64}{9} = \frac{72 - 64}{9} = \frac{8}{9}$

Binomial Distribution

If $X \sim B(n, p)$ (binomial distribution with $n$ trials and probability $p$ of success):

• $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ for $k = 0, 1, 2, \ldots, n$
• $E[X] = np$
• $\text{Var}(X) = np(1-p)$

Poisson Distribution

If $X \sim P_o(\lambda)$ (Poisson distribution with mean $\lambda$):

• $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$ for $k = 0, 1, 2, \ldots$
• $E[X] = \lambda$
• $\text{Var}(X) = \lambda$

In Tanzania, market vendors and small business owners use probability, expectation, and standard deviation to manage inventory and predict sales. For example, a shopkeeper in Dar es Salaam tracking daily sales of milk packets can calculate the expected number sold per day and the standard deviation to determine how much stock to keep. This helps avoid wastage (when too much milk expires unsold) and lost sales (when customers leave empty-handed), directly applying these statistical concepts to everyday business decisions and money management.