Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate probabilities in simple and compound events.
• Use probability to predict outcomes and assess risk.
• Evaluate the fairness of games of chance.
• Apply probability to real-life South African scenarios.
• Critically analyze statistical information about risk.

Lesson summary

Probability is a fundamental concept in mathematics that helps us understand and quantify uncertainty. In Mathematical Literacy, probability goes beyond just calculating numbers; it empowers you to make informed decisions in everyday life, especially when evaluating risks and predicting potential outcomes. Think about insurance, lottery games, predicting weather patterns affecting farming, or even understanding the spread of diseases. Being able to critically assess probabilities is crucial for making smart choices related to your finances, health, and 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, where 0 represents impossibility and 1 represents certainty.

Event: A specific outcome or set of outcomes we're interested in (e.g., rolling a 6 on a die).

Sample Space: The set of all possible outcomes (e.g., {1, 2, 3, 4, 5, 6} when rolling a die).

Probability Formula: P(Event) = (Number of favorable outcomes) / (Total number of possible outcomes)

Example 1: What is the probability of drawing a king from a standard deck of 52 cards?

Number of favorable outcomes: 4 (there are 4 kings in a deck)

Total number of possible outcomes: 52 (total number of cards) P(King) = 4/52 = 1/13 ≈ 0.077 or 7.7% This means there's approximately a 7.7% chance of drawing a king. 2.2 Compound Events: Compound events involve two or more events occurring together.

We differentiate between: Independent Events: The outcome of one event does not affect the outcome of the other event.

Example: Flipping a coin twice. P(A and B) = P(A)

P(B)

Dependent Events: The outcome of one event does affect the outcome of the other event.

Example: Drawing two cards from a deck without replacement. P(A and B) = P(A) P(B|A) where P(B|A) means "the probability of B given that A has already occurred." Example 2 (Independent Events): What is the probability of flipping a coin twice and getting heads both times? P(Heads on first flip) = 1/2 P(Heads on second flip) = 1/2 P(Heads and Heads) = (1/2) (1/2) = 1/4 = 0.25 or 25% Example 3 (Dependent Events): A bag contains 5 red balls and 3 blue balls. What is the probability of drawing two red balls in a row without replacing the first ball? P(Red on first draw) = 5/8 P(Red on second draw given that a red ball was drawn first) = 4/7 (since there are now only 4 red balls and 7 total balls left). P(Red and Red) = (5/8) (4/7) = 20/56 = 5/14 ≈ 0.357 or 35.7% 2.3 Predicting Outcomes: Probability can be used to predict how often an event will occur over a large number of trials. This is often done by multiplying the probability of the event by the number of trials.

Example 4: A fair coin is flipped 100 times. How many times would you expect to get heads? P(Heads) = 1/2 Number of trials = 100 Expected number of heads = (1/2) 100 = 50 Therefore, you would expect to get heads approximately 50 times. 2.4 Risk Assessment: Risk assessment involves identifying potential hazards and evaluating the likelihood and severity of their consequences. Probability plays a vital role in quantifying the likelihood part of this assessment.

Example 5: Insurance Insurance companies use probability to assess the risk of insuring individuals or assets. For instance, they calculate the probability of a car accident occurring based on factors like age, driving history, and location. The higher the probability of an accident, the higher the insurance premium.

Example 6: Predicting Loan Defaults Banks use probability to determine the likelihood of a borrower defaulting on a loan. They analyse factors such as credit score, income, and employment history to assess this risk. Higher default probabilities translate to higher interest rates charged on the loan. 2.5 Fairness of Games of Chance: A fair game of chance is one where each player has an equal probability of winning or losing. We can use probability to evaluate the fairness of a game. A game where the expected value is zero is considered fair. Expected Value = (Probability of winning Amount won) - (Probability of losing Amount lost)

Example 7: A lottery ticket costs R

5. The probability of winning R100 is 1/

1

0

0. Is this a fair game? P(Winning) = 1/100 Amount Won = R100 P(Losing) = 99/100 Amount Lost = R5 Expected Value = (1/100 R100) - (99/100 * R5) = R1 - R4.95 = -R3.95 Since the expected value is negative, this is not a fair game. On average, you are expected to lose R3.95 for every ticket you buy. Guided Practice (With Solutions)

Question 1: A spinner has 8 equal sections numbered 1 to

8. What is the probability of spinning an even number?

Solution: Favorable outcomes: {2, 4, 6, 8} (4 even numbers)

Total possible outcomes: {1, 2, 3, 4, 5, 6, 7, 8} (8 total numbers) P(Even number) = 4/8 = 1/2 = 0.5 or 50%

Commentary: This is a straightforward application of the basic probability formula. We identify the favorable outcomes (even numbers) and divide by the total possible outcomes.

Question 2: A bag contains 4 green marbles and 6 yellow marbles. Two marbles are drawn at random with replacement. What is the probability of drawing a green marble followed by a yellow marble?

Solution: P(Green on first draw) = 4/10 = 2/5 P(Yellow on second draw) = 6/10 = 3/5 (Since we replace the marble, the probabilities remain the same) P(Green and Yellow) = (2/5) (3/5) = 6/25 = 0.24 or 24%

Commentary: This involves independent events because replacing the first marble ensures the outcome of the first draw doesn't affect the second.

Question 3: A tuck shop sells sweets.