Probability
Data Handling and Probability · Probability
Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 5.0 Data Handling and Probability | Sub-Strand 5.2 Probability (6 Lessons)
Lesson objectives
By the end of this sub-strand, you should be able to:
- Determine the probability of a single event from equally likely outcomes.
- Distinguish between experimental and theoretical probability.
- Apply the addition and multiplication rules to combined events.
- Represent outcomes using tables, lists and tree diagrams.
Probability
Will it rain this afternoon? Will the coin land heads? We cannot know for certain, but we can measure how likely something is.
That measurement is probability, and it is always a number between 0 and 1.
a) The probability scale
Every probability sits somewhere on a scale from impossible to certain.
A probability of 0 means the event cannot happen. A probability of 1 means it must happen. Halfway along, at 0.5, the event is as likely as not.
No probability is ever less than 0 or more than 1. An answer of 1.4 or −0.2 means something has gone wrong.
b) Theoretical probability
When all outcomes are equally likely, count them.
Divide the number of outcomes you want by the total number of outcomes.
Notice the last line. The probability of something happening plus the probability of it not happening always adds to 1. That gives you a quick way to find one from the other.
c) Experimental probability
Sometimes outcomes are not equally likely, or you cannot count them. A bottle top does not land flat or upside down with equal chance.
Then you carry out a trial. Toss the top many times and record what happens.
Experimental probability is the number of successes divided by the number of trials.
The more trials you run, the closer the experimental value creeps towards the true one. Ten tosses tell you little. Five hundred tell you a lot.
d) Combined events
When two things happen one after the other, a tree diagram keeps track of every possible outcome.
Each path from left to right is one outcome. Tossing a coin twice gives four paths, and each is equally likely.
e) Reading a tree diagram
Two rules cover every question.
Multiply along a path to find the probability of that whole path. Add between paths when more than one outcome satisfies the question.
Remember the words. "And" means multiply. "Or" means add.
Check your work by adding all the end probabilities. They must come to exactly 1.
f) Where this is used
An insurer prices cover using probability of claims. A weather forecaster gives rain a percentage chance. A farmer choosing a planting date weighs the chance of the rains arriving on time.
Words to know
- Probability -- a measure of how likely an event is, taking a value from 0 to 1.
- Mutually exclusive events -- events that cannot both occur at the same time.
- Independent events -- events where the outcome of one does not affect the other.
- Complement -- the outcome that the event does not occur, with probability 1 minus the event probability.
- Experimental probability -- the proportion of successes observed over a number of actual trials.
:::checkpoint Check yourself
- A bag holds 4 blue and 6 green beads. What is the probability of picking blue?
- If P(win) is 0.3, what is P(not win)?
- A coin is tossed twice. What is the probability of two tails?
- In a tree diagram, when do you multiply and when do you add? :::
Bridge to practice
The exercises begin with single-event probability and complements, move through the addition and multiplication rules, and finish with without-replacement problems and the reasoning behind common misconceptions. For any question involving two selections, ask first whether the item was replaced, since that single detail changes the denominator and therefore the whole answer.