Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate the probability of simple and compound events.
• Use probability to predict the likelihood of real-world events.
• Interpret probabilities as fractions, decimals, and percentages.
• Understand and apply the concepts of risk and relative risk.
• Compare theoretical and experimental probability.

Lesson summary

This week, we delve into the exciting world of probability, focusing on how we can use it to predict outcomes and understand risk. Probability isn't just a theoretical concept; it's a powerful tool that influences many aspects of our lives in South Africa, from deciding whether to buy insurance to understanding the chances of winning the Lotto, or even assessing the risks associated with driving in different weather conditions. Understanding probability allows us to make more informed decisions and better assess the potential consequences of our actions. This is crucial for financial literacy, responsible citizenship, and personal safety.

Lesson notes

2.1 Basic Probability: Probability is a measure of the likelihood that an event will occur. It's expressed as a number between 0 and 1 (inclusive), where 0 means the event is impossible, and 1 means the event is certain. Probability can also be represented as a percentage (0% to 100%).

The basic formula for probability is: ``` Probability of an event (P(A)) = (Number of favorable outcomes) / (Total number of possible outcomes) ``` Event: A specific outcome or set of outcomes.

Favorable outcome: An outcome that satisfies the conditions of the event.

Total number of possible outcomes: All possible outcomes of the situation.

Example 1: A Fair Die What is the probability of rolling a 4 on a fair six-sided die?

Favorable outcome: Rolling a 4 (1 outcome)

Total possible outcomes: Rolling a 1, 2, 3, 4, 5, or 6 (6 outcomes) P(rolling a 4) = 1/6 As a percentage, this is approximately 16.67%.

Example 2: Choosing a Learner at Random In a class of 30 learners, 12 are boys and 18 are girls. What is the probability of randomly selecting a girl?

Favorable outcome: Selecting a girl (18 outcomes)

Total possible outcomes: Selecting any learner (30 outcomes) P(selecting a girl) = 18/30 = 3/5 = 0.6 = 60% 2.2 Compound Events: Compound events involve two or more events occurring together.

There are two main types: Independent Events: The outcome of one event does not affect the outcome of the other.

Example: Flipping a coin and then rolling a die.

Formula: P(A and B) = P(A)

P(B)

Dependent Events: The outcome of one event does affect the outcome of the other. This is often seen when sampling without replacement.

Example: Drawing two cards from a deck without replacing the first card.

Formula: P(A and B) = P(A) P(B|A) (where P(B|A) is the probability of B given that A has already occurred).

Example 3: Independent Events - Coin and Die What is the probability of flipping a coin and getting heads, and rolling a 6 on a die? P(Heads) = 1/2 P(Rolling a 6) = 1/6 P(Heads and Rolling a 6) = (1/2) (1/6) = 1/12 Example 4: Dependent Events - Drawing Cards A box contains 5 red balls and 3 blue balls. You draw two balls without replacement. What is the probability of drawing a red ball first, followed by another red ball? P(First ball is red) = 5/8 P(Second ball is red, given the first was red) = 4/7 (since there are now only 4 red balls and 7 total balls left) P(Red then Red) = (5/8) (4/7) = 20/56 = 5/14 2.3 Risk and Relative Risk: Risk: The possibility of suffering harm or loss. In probability terms, risk is often associated with the probability of an undesirable event occurring.

Relative Risk: A comparison of the risk in two different groups. It's calculated as the ratio of the probability of an event occurring in one group to the probability of the event occurring in another group.

Formula: Relative Risk = (Probability of event in group A) / (Probability of event in group B)

Example 5: Understanding Risk - Road Accidents Suppose statistics show that 1 in 500 drivers in Gauteng are involved in a car accident each year, compared to 1 in 800 drivers in the Western Cape. Which province carries the greater risk?

Gauteng accident probability: 1/500 = 0.002 Western Cape accident probability: 1/800 = 0.00125 Drivers in Gauteng have a higher risk of being involved in an accident.

Example 6: Calculating Relative Risk Using the accident probabilities from Example 5, what is the relative risk of driving in Gauteng compared to the Western Cape? Relative Risk = (1/500) / (1/800) = (800/500) = 1.6 This means the risk of being in an accident is 1.6 times higher in Gauteng than in the Western Cape. 2.4 Theoretical vs.

Experimental Probability: Theoretical Probability: The probability calculated based on mathematical reasoning and assumptions about the event (e.g., a fair die).

Experimental Probability: The probability calculated based on the results of an experiment or observation.

It's calculated as: Experimental Probability = (Number of times the event occurs) / (Total number of trials) As the number of trials increases, the experimental probability should get closer to the theoretical probability (Law of Large Numbers).

Example 7: Coin Toss Simulation You flip a coin 100 times and get heads 55 times.

Theoretical probability of heads: 1/2 = 0.5 Experimental probability of heads: 55/100 = 0.55 The experimental probability is close to the theoretical probability. If you flipped the coin 1000 times, the experimental probability would likely be even closer to 0.

5. Guided Practice (With Solutions)

Question 1: A bag contains 7 yellow sweets, 5 green sweets, and 8 red sweets. If you pick a sweet at random, what is the probability that it is green?

Solution: Total number of sweets: 7 + 5 + 8 = 20 Number of green sweets: 5 P(Green sweet) = 5/20 = 1/4 = 0.25 = 25%

Commentary: This is a straightforward application of the basic probability formula. We identified the favorable outcome (green sweet) and divided it by the total possible outcomes (all sweets).