25
PROBABILITY
CONTENTS
Probabilities 225
Basic Relationships of Probability 228
Complement 228
Union and Intersection 228
Mutually Exclusive Events 228
Addition Law 228
Conditional Probability 230
The Multiplication Law 230
Tree Diagrams 233
Bayes’ Theorem 235
Probability is a measure of the likelihood that an event will occur, represented
on a scale from 0 to 1. A probability close to 0 suggests unlikelihood, while a
probability close to 1 suggests near certainty (Figure 25.1). Probability is a
mathematical tool used to study and predict outcomes, such as the chance of rain,
winning a bet, exam success, or financial gains like increased share prices or
return on investment (ROI).
Figure 25.1 Probability Scale
NOTE:
• An experiment is a process or procedure that generates outcomes (e.g.,
flipping a coin, rolling dice, conducting surveys, or running scientific trials).
• Events are specific outcomes from a probability experiment, usually denoted
by capital letters (e.g., A). Examples of events include rolling a six on a die,
drawing a red card from a deck of cards, or getting heads when flipping a
coin.
• The probability of an event (A) is represented as 𝑃(𝐴).
• The probability of an event not occurring (AC) is found by subtracting the
probability of A from 1: 1−𝑃(𝐴).
PROBABILITIES
There are three approaches of assigning probabilities: classical, relative
frequency, and subjective methods.
Method Explanation
Classical The classical method is appropriate when all the outcomes are
method equally likely to occur. If N outcomes are possible, a probability
of 1/N is assigned to each outcome.
226 Olaniyi Evans | University Mathematics
An example of the classical method is rolling a die because it is
equally likely that you will land on any of the 6 numbers on the
die: 1, 2, 3, 4, 5, or 6. Another example is a coin toss because it is
equally likely that your toss will yield a heads or tails.
Relative The relative frequency method is appropriate when an
frequency experiment is repeated a large number of times and a particular
outcome occurs a percentage of the time. Then that particular
percentage is the probability of that outcome.
For example, if a company produces 100,000 phones in a year,
and 1,000 of those phones are defective, the probability of that
company producing a defective phone is approximately 1,000
out of 100,000, or 0.01.
The probability of an event occurring is therefore given by:
𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐬𝐮𝐜𝐜𝐞𝐬𝐬𝐟𝐮𝐥 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬
𝑷(𝐞𝐯𝐞𝐧𝐭) =
𝐓𝐨𝐭𝐚𝐥 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐩𝐨𝐬𝐬𝐢𝐛𝐥𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬
Subjective The subjective method is based on opinion, previous experience
method or intuition. After considering all available information, a
probability value that shows your degree of belief (on a scale
from 0 to 1) that the outcome will occur is specified.
Irrespective of the method used, there are two basic requirements for assigning
probabilities:
a) The probability assigned to each outcome must be between 0 and 1.
b) The sum of the probabilities for all the outcomes must equal 1.
EXAMPLE 25.1
Consider the toss of a coin; the two outcomes are head and tail. What is the
probability for each outcome?
S O L U T I O N tips
The two outcomes are equally likely: classical probability.
Thus, the probability of observing a head is ½ (or 0.50).
Similarly, the probability of observing a tail is also ½ (or 0.50).
EXAMPLE 25.2
A student records the number of his designer accessories and obtains the
following results.
Designer Label Gucci Prada Chanel Burbery Rolex Coach
Accessories 2 10 7 8 2 1
Suppose the student wants to randomly choose one of these accessories to wear.
What is the probability that the accessory chosen will be (a) Prada (b) Rolex
S O L U T I O N tips
a) By the relative frequency method, the probability of choosing Prada is
Number of Prada accessories 10 1
𝑃(Prada) = = = = 0.33
Total number of accessories 30 3
b) The probability of choosing Rolex is
Number of Rolex accessories 2 1
𝑃(Rolex) = = = = 0.07
Total number of accessories 30 15
EXAMPLE 25.3
What is the probability of rolling a sum of 10 with two dice?
Chapter 25| Probability 227
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