Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Define basic probability terms (event, outcome, sample space).
• Calculate the probability of a single event.
• Express probabilities as fractions, decimals, and percentages.
• Analyse simple probability experiments (like coin tosses).
• Understand the difference between theoretical and experimental probability.

Lesson summary

Probability is a fundamental concept that helps us understand the likelihood of events happening. It's all around us, from predicting the weather forecast to understanding the chances of winning the lottery. In the South African context, probability helps us make informed decisions about everything from farming practices (predicting rainfall) to financial investments (assessing risk) and even health outcomes (understanding the spread of diseases). It allows us to quantify uncertainty and make better judgments based on available information. This week, we'll focus on the basic concepts and simple experiments to build a strong foundation in probability.

Lesson notes

2.1 Basic Terminology: Experiment: An activity involving chance that leads to results. Examples include tossing a coin, rolling a die, or drawing a card from a deck.

Outcome: A possible result of an experiment. For example, when tossing a coin, the possible outcomes are "Heads" or "Tails." Sample Space: The set of all possible outcomes of an experiment. We often denote the sample space by the letter 'S'. For example, the sample space for rolling a six-sided die is S = {1, 2, 3, 4, 5, 6}.

Event: A specific outcome or set of outcomes in which we are interested. For example, rolling an even number on a die is an event. It consists of the outcomes {2, 4, 6}.

Probability: A measure of how likely an event is to occur. It is a number between 0 and 1 (inclusive), where 0 means the event is impossible and 1 means the event is certain. We denote the probability of an event 'A' by P(A). 2.2 Calculating Probability: The basic formula for calculating the probability of an event is: P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes) 2.3 Representing Probabilities: Probabilities can be expressed in three ways: Fractions: e.g., 1/2, 3/4, 5/6 Decimals: e.g., 0.5, 0.75, 0.833 Percentages: e.g., 50%, 75%, 83.3% Remember that to convert a fraction to a decimal, you divide the numerator by the denominator. To convert a decimal to a percentage, you multiply by 100. 2.4 Theoretical vs.

Experimental Probability: Theoretical Probability: The probability of an event based on mathematical calculations and assumptions about the experiment. It's what should happen in an ideal situation. For example, the theoretical probability of rolling a '4' on a fair six-sided die is 1/

6. Experimental Probability: The probability of an event based on the results of actual experiments. It's what actually happens when you repeat the experiment many times. For example, if you roll a die 60 times and get a '4' only 8 times, the experimental probability of rolling a '4' is 8/

6

0. The more times you repeat an experiment, the closer the experimental probability tends to get to the theoretical probability. This is known as the Law of Large Numbers. 2.5 Worked

Examples: Example 1: Tossing a Coin What is the probability of getting Heads when tossing a fair coin?

Experiment: Tossing a coin Sample Space: S = {Heads, Tails} Event: Getting Heads Number of favourable outcomes: 1 (Heads)

Total number of possible outcomes: 2 (Heads or Tails) Therefore, P(Heads) = 1/2 = 0.5 = 50% Example 2: Rolling a Die What is the probability of rolling an even number on a fair six-sided die?

Experiment: Rolling a die Sample Space: S = {1, 2, 3, 4, 5, 6} Event: Rolling an even number Favourable outcomes: {2, 4, 6} Number of favourable outcomes: 3 Total number of possible outcomes: 6 Therefore, P(Even number) = 3/6 = 1/2 = 0.5 = 50% Example 3: Drawing a Card A standard deck of cards has 52 cards, with 4 suits (Hearts, Diamonds, Clubs, Spades) and 13 cards in each suit (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King). What is the probability of drawing a Heart?

Experiment: Drawing a card Sample Space: 52 cards Event: Drawing a Heart Number of favourable outcomes: 13 (there are 13 Hearts)

Total number of possible outcomes: 52 Therefore, P(Heart) = 13/52 = 1/4 = 0.25 = 25% Example 4: Marble Selection A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of selecting a blue marble at random?

Experiment: Selecting a marble Sample Space: 5 red + 3 blue + 2 green = 10 marbles Event: Selecting a blue marble Number of favourable outcomes: 3 (there are 3 blue marbles)

Total number of possible outcomes: 10 Therefore, P(Blue Marble) = 3/10 = 0.3 = 30% Guided Practice (With Solutions)

Question 1: A spinner is divided into 8 equal sections, numbered 1 to

8. What is the probability of the spinner landing on an odd number?

Solution: Experiment: Spinning the spinner Sample Space: S = {1, 2, 3, 4, 5, 6, 7, 8} Event: Landing on an odd number Favourable outcomes: {1, 3, 5, 7} Number of favourable outcomes: 4 Total number of possible outcomes: 8 P(Odd number) = 4/8 = 1/2 = 0.5 = 50%

Commentary: We identified the sample space and the event of interest. Then, we counted the number of favourable outcomes and divided it by the total number of possible outcomes to find the probability.

Question 2: A box contains 20 coloured pencils. 6 are red, 8 are blue and 6 are yellow. If you pick a pencil at random, what is the probability that it is NOT blue?

Solution: Experiment: Picking a pencil Sample Space: 20 pencils Event: Picking a pencil that is NOT blue.

Favourable outcomes: Red pencils + Yellow pencils = 6 + 6 = 12 Total number of possible outcomes: 20 P(Not Blue) = 12/20 = 3/5 = 0.6 = 60%

Commentary: It is important to read the question carefully. This question asked for the probability of not selecting a blue pencil. We needed to calculate the number of pencils that were not blue.

Question 3: You roll a standard six-sided die twice.