2.1 Basic Axioms

1. For any event $E$: $0 \leq P(E) \leq 1$
2. For the sample space $S$: $P(S) = 1$
3. For mutually exclusive events $E_1, E_2, \ldots$: $P(E_1 \cup E_2 \cup \cdots) = P(E_1) + P(E_2) + \cdots$

2.2 Key Theorems

• $P(\emptyset) = 0$ (impossible event)
• $P(E') = 1 - P(E)$ (complement)
• For any two events $A$ and $B$: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

2.3 Probability of an Event

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

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

Solution: $S = \{1,2,3,4,5,6\}$, $n(S) = 6$ Even numbers: $\{2,4,6\}$, $n(E) = 3$ $P(E) = \frac{3}{6} = 0.5$

3.1 Independent Events

Two events are independent if the occurrence of one does not affect the occurrence of the other. For independent events $A$ and $B$: $P(A \cap B) = P(A) \times P(B)$

Example: A box contains 5 blue and 6 black balls. A ball is drawn and replaced before a second draw. Find the probability that both balls drawn are black.

Solution: $P(\text{black first}) = \frac{6}{11}$, $P(\text{black second}) = \frac{6}{11}$ $P(\text{both black}) = \frac{6}{11} \times \frac{6}{11} = \frac{36}{121}$

3.2 Dependent Events (Conditional Probability)

When events are dependent, we use conditional probability: $P(B|A) = \frac{P(A \cap B)}{P(A)}$

This reads as "probability of $B$ given that $A$ has occurred."

Example: A fair die is rolled once and an odd number appears. Find the probability that it is a prime number.

Solution: Given odd number occurred: $A = \{1,3,5\}$, $P(A) = \frac{3}{6}$ Prime numbers in $A$: $\{3,5\}$, $P(A \cap B) = \frac{2}{6}$ $P(B|A) = \frac{2/6}{3/6} = \frac{2}{3}$

4.1 Discrete Random Variables

A discrete random variable takes countable values. A probability distribution assigns probabilities to each value such that $\sum P(X = x) = 1$.

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

Solution: Sample space: $HH, HT, TH, TT$

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

4.2 Expected Value and Variance

The expected value (mean) of a discrete random variable: $E[X] = \sum x_i p_i$

The variance: $\text{Var}[X] = E[X^2] - (E[X])^2$ where $E[X^2] = \sum x_i^2 p_i$

Example: Find the mean of the distribution where $P(X=1) = 0.1$, $P(X=2) = 0.3$, $P(X=3) = 0.2$, $P(X=4) = 0.3$, $P(X=5) = 0.1$

Solution: $E[X] = 1(0.1) + 2(0.3) + 3(0.2) + 4(0.3) + 5(0.1) = 3$

5.1 Binomial Distribution

Used when there are $n$ independent trials, each with probability $p$ of success. $P(X = k) = {}^nC_k p^k (1-p)^{n-k}$ Denoted $X \sim B(n, p)$, with $E[X] = np$ and $\text{Var}[X] = np(1-p)$

5.2 Poisson Distribution

Used for counting events that occur randomly in a fixed interval (time or space). Given mean $\lambda$: $P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$ Denoted $X \sim P_o(\lambda)$, with $E[X] = \lambda$ and $\text{Var}[X] = \lambda$

5.3 Normal Distribution

A continuous distribution with mean $\mu$ and standard deviation $\sigma$, denoted $X \sim N(\mu, \sigma^2)$. The standard normal distribution has $\mu = 0$ and $\sigma = 1$.

To standardize: $z = \frac{x - \mu}{\sigma}$

In Tanzania, probability is used in everyday situations such as predicting weather patterns for farming decisions. For example, if the Tanzania Meteorological Authority forecasts a 70% chance of rain, a small-scale farmer in Morogoro can use this probability to decide whether to plant crops that require rainfall or to delay planting. Similarly, mobile money service providers use probability distributions to predict the number of transactions per hour at markets like Mwenge, helping them ensure enough float is available to serve customers.