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Section 5.2 Probability
Probability refers to the likelihood or chance of an event occurring.
It is a numerical measure that quantifies how likely it is that a specific outcome will happen in a situation with uncertain results.
Probabilities are expressed as numbers between 0 and 1 or as percentages from 0% to 100%.
Subsection 5.2.1 Introduction to Probability
Activity 5.2.1 .
\(\textbf{Work in groups}\)
\(\textbf{Classroom Experiment: Tossing a Coin} \)
\(\textbf{Materials: One coin per pair of learners.} \)
Each pair of students tosses a coin 20 times.
Record how many times it lands on Heads and how many times on Tails.
Complete a tally table and discuss:
Was any outcome more likely?
\(\textbf{Key Takeaway}\)
Understanding probability helps us predict outcomes and make decisions in real life (e.g., weather forecasts, games of chance, insurance, business).
Example 5.2.1 .
A coin is tossed once. What is the probability of getting a Head?
Solution .
Sample space = {Head, Tail}
Number of favorable outcomes (Head) = 1
\begin{gather*}
\textbf{P(Head)} = \frac{1}{2} = 0.5
\end{gather*}
Subsection 5.2.2 Range of Probability
Activity 5.2.2 .
\(\textbf{Work individually}\)
Draw a horizontal line labeled from 0 to 1.
Place the following events on the line where you think they belong:
The sun will rise tomorrow
Rolling a 3 on a fair die
Picking a red ball from a bag with 1 red and 9 blue balls
\(\textbf{Key Takeaway}\)
Probability is written as a number between 0 and 1
\(0.5\) = Event is equally likely to happen or not happen
Example 5.2.2 .
A bag contains 6 red and 2 blue marbles. What is the probability of drawing a red marble?
Solution .
Total marbles
\(= 6 + 2 = 8\)
Favorable outcomes (red)
\(= 6\)
\begin{gather*}
\textbf{P(Red)} = \frac{6}{8} = 0.75
\end{gather*}
\begin{gather*}
= 75%
\end{gather*}
This is a likely event, since
\(0.75 \) is closer to 1.
Subsection 5.2.3 Exclusive Events
Two or more events are mutually exclusive if they cannot happen at the same time.
Activity 5.2.3 .
\(\textbf{Work in groups}\)
Event A: Getting an even number
Event B: Getting an odd number
Discuss:
Are these events mutually exclusive?
what is
\(\textbf{P(even) + P(odd)?} \)
\(\textbf{Key Takeaway}\)
In other words, the occurrence of one event means the other cannot occur.
Example 5.2.3 .
A bag contains 3 red, 2 blue, and 1 yellow marble. One marble is drawn.
Find the probability of drawing:
Solution .
\(\textbf{P(Red)} = \frac{3}{6} \textbf{and} \textbf{ P(Yellow)} = \frac{1}{6}\)
Red and yellow are mutually exclusive (you can’t pick both at once)
\begin{gather*}
\textbf{P(Red or Yellow)}
\end{gather*}
\begin{gather*}
\frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}
\end{gather*}
Subsection 5.2.4 Independent Events
Two events are independent if the outcome of one does not affect the outcome of the other.
Activity 5.2.4 .
\(\textbf{Work in groups}\)
Toss a coin and roll a die at the same time.
Record the outcome (e.g., Heads and 4).
Discuss:
Are these two events independent?
Can the coin toss result affect the die roll?
\(\textbf{Key Takeaway}\)
In other words, the result of one event has no influence on the second event.
Example 5.2.4 .
A coin is tossed twice. What is the probability of getting heads both times?
Solution .
First toss:
\(\textbf{P(Heads)} = 1/2 \)
Second toss:
\(\textbf{P(Heads)} = 1/2 \)
They are independent, so:
\begin{gather*}
\textbf{P(Heads and Heads)} = \frac{1}{2} × \frac{1}{2}
\end{gather*}
\begin{gather*}
= \frac{1}{4}
\end{gather*}
Subsection 5.2.5 Tree Diagram
A tree diagram is a visual tool used to show all possible outcomes of two or more events, especially when calculating probabilities of combined or successive events.
Activity 5.2.5 .
\(\textbf{Work in groups}\)
Construct a tree diagram for tossing a coin twice.
Label the branches with:
First toss: H or T each with
\(\frac{1}{2} \)
Second toss: H or T again each with
\(\frac{1}{2} \)
Find the probability of each e.g(P(HH), P(HT))
\(\textbf{Key Takeaway}\)
It helps organize outcomes in branches and is particularly useful for independent events.
Example 5.2.5 .
A bag has 2 red and 1 green marble. A marble is drawn, replaced, then another is drawn.
Draw a tree diagram and find the probability of getting
One red and one green (in any order)
Solution .
\(\textbf{P(Red)} = \frac{2}{3} \)
\(\textbf{P(Green)} = \frac{1}{3}\)
Tree Diagram where R = P(Red) and G = P(Green)
\begin{gather*}
= \frac{2}{3} × \frac{2}{3} = \frac{4}{9}
\end{gather*}
\begin{gather*}
= \frac{2}{3} × \frac{1}{3} = \frac{2}{9}
\end{gather*}
\begin{gather*}
= \frac{1}{3} × \frac{2}{3} = \frac{2}{9}
\end{gather*}
Total
\begin{gather*}
= \frac{2}{9} + \frac{2}{9}
\end{gather*}
Subsection 5.2.6 Application of Probability
Probability helps us make informed decisions in daily life, games, weather forecasting, business, health and risk analysis.
Activity 5.2.6 .
\(\textbf{Work in groups}\)
Look at your class register for the past 10 school days.
Count how many days a specific student was present.
Estimate the probability that the student will attend tomorrow.
\(\textbf{Key Takeaway}\)
Probability allows us to predict, estimate risk, and plan ahead based on chance and likelihood.
Example 5.2.6 .
A weather report says there’s a 70% chance of rain tomorrow.
This means the event is not certain, but it is likely.
Example 5.2.7 .
A study shows that 1 in 4 people are likely to catch a cold during the cold season.
\(\textbf{P(Catch a cold) = }\)
\begin{gather*}
\frac{1}{4} = 0.25
\end{gather*}
\begin{gather*}
\textbf{The probability of one catching cold out of four is 0.25 which is 25%}
\end{gather*}
