Probability of an event

The probability of an event is the number of ways an event can occur divided by the total number of possible outcomes. A set of all possible outcomes is called a sample space (S), while a specified outcome is called an event (E). The probability of an event measures how likely it is to happen. It is denoted by P(E) and read as "the probability that event E occurs." It is given by:

$P(E) = \frac{\text{Number of ways event E can occur}}{\text{Total number of possible outcomes}}$

$P(E) = \frac{n(E)}{n(S)}$

Probabilities are always numbers between 0 and 1 ($0 \leq P(E) \leq 1$) and can be expressed as fractions, decimals, or percentages. The closer the probability of an event is to 0, the less likely the event is to occur. An event with a probability of 0 can never happen. The closer the probability of an event is to 1, the more likely the event is to occur. An event with a probability of 1 is certain to happen.

Example 1

A die is rolled once. What is the probability of getting:

a) an even number

b) a number greater than 4

Solution:

The sample space after rolling a die is: $S = \{1, 2, 3, 4, 5, 6\}$, $n(S) = 6$

a) Let A represent an even number: $A = \{2, 4, 6\}$, $n(A) = 3$

$P(A) = \frac{n(A)}{n(S)} = \frac{3}{6} = \frac{1}{2}$

The probability of an even number is 1/2.

b) Let E represent a number greater than 4: $E = \{5, 6\}$, $n(E) = 2$

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

The probability of a number greater than 4 is 1/3.

A Tsh 100 coin has two sides (head and antelope). A Tsh 200 coin has head and lions. A Tsh 500 coin has head (H) and an animal side (T). When tossing a coin, the expected outcomes are either:

1. getting head up (H)
2. getting an animal up (T)

So, $S = \{H, T\}$. Tossing a British penny gives the same result: $\{H, T\}$. Tossing a coin has two expected outcomes: Head (H) or Tail (T).

Example 2

A coin is tossed twice. What is the probability of getting two tails?

Solution:

Sample space $= \{HH, HT, TH, TT\}$ and event $= \{TT\}$

$\text{Probability of getting two tails} = \frac{\text{Number of two tails outcomes}}{\text{Total number of sample space outcomes}} = \frac{1}{4}$