Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Calculate probability for independent events.
• Calculate probability for dependent events.
• Distinguish between independent and dependent events.
• Apply probability to assess everyday risks.
• Critically evaluate probability statements in media.

Lesson summary

Probability is a fundamental concept that governs many aspects of our lives. Understanding combined events and assessing everyday risks allows us to make more informed decisions, from purchasing insurance to interpreting health statistics to understanding the odds of winning the lottery. In the South African context, where socio-economic factors often intersect with unpredictable events (like weather patterns affecting agriculture or economic fluctuations impacting job security), a solid grasp of probability is crucial for navigating uncertainty and making responsible choices.

Lesson notes

2.1 Independent Events: Two events are considered independent if the outcome of one event does not affect the outcome of the other. The probability of two independent events, A and B, both occurring is the product of their individual probabilities: P(A and B) = P(A) * P(B)

Example 1: Coin Toss and Dice Roll Imagine flipping a fair coin and rolling a fair six-sided die. The outcome of the coin toss (Heads or Tails) does not influence the outcome of the die roll (1, 2, 3, 4, 5, or 6). What is the probability of getting Heads on the coin AND rolling a 4 on the die? P(Heads) = 1/2 P(Rolling a 4) = 1/6 P(Heads and Rolling a 4) = (1/2) (1/6) = 1/12 Example 2: Drawing Cards (with Replacement) A standard deck of 52 cards is used. A card is drawn, then replaced, and then another card is drawn. What is the probability of drawing a King, replacing it, and then drawing a Queen? P(King) = 4/52 = 1/13 P(Queen) = 4/52 = 1/13 P(King and Queen) = (1/13) (1/13) = 1/169 2.2 Dependent Events: Two events are dependent if the outcome of the first event does affect the outcome of the second event. The probability of two dependent events, A and B, both occurring is: P(A and B) = P(A) * P(B|A) Where P(B|A) represents the conditional probability of event B occurring, given that event A has already occurred.

Example 3: Drawing Cards (without Replacement) A standard deck of 52 cards is used. A card is drawn and not replaced. What is the probability of drawing a King, and then drawing a Queen? P(King) = 4/52 = 1/13 P(Queen | King has already been drawn) = 4/51 (Because there are now only 51 cards left in the deck) P(King and Queen) = (1/13) (4/51) = 4/663 Example 4: Traffic Light Scenario Let's say you are driving to school. The probability of hitting a green light at the first intersection is 0.

7. If you hit a green light at the first intersection, the probability of hitting a green light at the second intersection is 0.8 (perhaps the traffic lights are synchronized). What is the probability of hitting green lights at both intersections? P(Green Light 1) = 0.7 P(Green Light 2 | Green Light 1) = 0.8 P(Green Light 1 and Green Light 2) = 0.7 0.8 = 0.56 2.3 Everyday Risk: Probability concepts are crucial for assessing risks in daily life.

Consider the following: Insurance: Insurance companies use probability to determine premiums. They assess the likelihood of an event occurring (e.g., car accident, house fire) based on historical data and individual risk factors. Higher probability events lead to higher premiums.

Financial Investments: Investing in the stock market involves risk. Different investments have different probabilities of success or failure. Understanding these probabilities helps investors make informed decisions about where to allocate their money.

Health Risks: Doctors use probability to estimate the likelihood of developing certain diseases based on factors like family history, lifestyle, and environmental exposure. These probabilities inform decisions about preventative measures and treatment options.

Lottery: Understanding the very low probability of winning the lottery can help people make informed decisions about how they spend their money. While the payout is large, the probability is so low that it's usually a poor financial decision to purchase lottery tickets regularly. Guided Practice (With Solutions)

Question 1: A bag contains 5 red marbles and 3 blue marbles. A marble is drawn, and then replaced. Then, another marble is drawn. What is the probability of drawing a red marble followed by a blue marble?

Solution: This is an example of independent events because the first marble is replaced. P(Red) = 5/8 P(Blue) = 3/8 P(Red and Blue) = (5/8) (3/8) = 15/64 Question 2: A group of students consists of 12 boys and 8 girls. Two students are randomly selected to represent the class at a school event. What is the probability that both students selected are girls?

Solution: This is an example of dependent events because the first student selected is not replaced. P(First student is a girl) = 8/20 = 2/5 P(Second student is a girl | First student was a girl) = 7/19 (Now there are only 7 girls and 19 total students left) P(Both students are girls) = (2/5) (7/19) = 14/95 Question 3: In a survey, it was found that 60% of adults in a town own a car, and 40% own a smartphone. If 25% of adults own both a car and a smartphone, what is the probability that a randomly selected adult owns a smartphone, given that they own a car?

Solution: This is a conditional probability problem. We're looking for P(Smartphone | Car). P(Car) = 0.60 P(Car and Smartphone) = 0.25 Using the formula: P(Smartphone | Car) = P(Car and Smartphone) / P(Car) P(Smartphone | Car) = 0.25 / 0.60 = 5/12 ≈ 0.4167 or 41.67% Independent Practice (Questions Only) A weather forecast predicts a 70% chance of rain on Saturday and a 60% chance of rain on Sunday. Assuming the events are independent, what is the probability that it will rain on both Saturday and Sunday?