Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate probability for simple and compound events.
• Interpret probabilities as measures of risk.
• Use probability to predict outcomes.
• Distinguish between independent and dependent events.
• Analyse data to determine probabilities.

Lesson summary

Probability is a fundamental concept in Mathematical Literacy, and understanding it is crucial for making informed decisions in various aspects of life. In the South African context, grasping probability helps us understand risks associated with investments, gambling (e.g., Lotto, sports betting), health (e.g., risks of certain diseases), and even weather patterns affecting agriculture. This week, we will focus on predicting outcomes and assessing risk using probability. Specifically, we will explore how to calculate probabilities, interpret their meaning, and apply them to real-world scenarios.

Lesson notes

2.1 Basic Probability Probability is the measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The probability of an event 'A' is denoted as P(A).

Formula: P(A) = Number of favorable outcomes / Total number of possible outcomes Example 1: A bag contains 5 red marbles and 3 blue marbles. What is the probability of selecting a red marble? Number of favorable outcomes (red marbles) = 5 Total number of possible outcomes (total marbles) = 5 + 3 = 8 P(Red) = 5/8 = 0.625 or 62.5% Example 2: A fair six-sided die is rolled. What is the probability of rolling a 4? Number of favorable outcomes (rolling a 4) = 1 Total number of possible outcomes (possible numbers on a die) = 6 P(4) = 1/6 = 0.1667 or approximately 16.67% 2.2 Compound Events Compound events involve two or more events happening together.

There are two main types: independent and dependent events.

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

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

P(B)

Example 3: You flip a coin and then roll a die. What is the probability of getting heads on the coin and a 3 on the die? P(Heads) = 1/2 P(3) = 1/6 Since the events are independent: P(Heads and 3) = (1/2) (1/6) = 1/12 Dependent Events: The outcome of one event does affect the outcome of the other. This usually happens when you're drawing items without replacement.

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

Example 4: A bag contains 4 green balls and 3 yellow balls. You draw a ball, don't replace it, and then draw another ball. What is the probability that both balls are green? P(First ball is green) = 4/7 Given the first ball was green, there are now only 3 green balls and 3 yellow balls left, making a total of

6. So, P(Second ball is green, given the first was green) = 3/6 = 1/2 P(Both balls are green) = (4/7) (1/2) = 2/7 2.3 Probability Tree Diagrams Tree diagrams are a visual way to represent and calculate probabilities, especially for compound events. Each branch represents a possible outcome, and the probabilities are written along the branches.

Example 5: A weather forecast predicts a 60% chance of rain on Saturday and a 70% chance of rain on Sunday. Draw a tree diagram and calculate the probability that it will rain on both days.

Saturday: Rain (R): P(R) = 0.6 No Rain (NR): P(NR) = 1 - 0.6 = 0.4 Sunday (dependent on Saturday):

Note: Because we are not told whether the chance of rain on Sunday is dependent on Saturday's weather, we assume that it is an independent event, although in reality weather patterns are dependent. This is for simplification.

Rain (R): P(R) = 0.7 No Rain (NR): P(NR) = 1 - 0.7 = 0.3 Tree Diagram: ``` Saturday Sunday Probability / \ R (0.6) NR (0.4) / \ / \ R(0.7) NR(0.3) R(0.7) NR(0.3) RR (0.60.7) RN(0.60.3) NR(0.40.7) NN(0.40.3) ``` P(Rain on both days) = P(RR) = 0.6 0.7 = 0.42 or 42% 2.4 Predicting Outcomes and Risk Probability allows us to make predictions about future events. A higher probability suggests a higher likelihood of the event occurring. Risk is often assessed by considering the probability of an undesirable outcome and its potential consequences.

Example 6: A medical study shows that a new drug has a 90% success rate in treating a certain disease. This means that for every 100 patients treated with the drug, approximately 90 are expected to recover.

However, there is still a 10% risk of the drug not working.

Example 7 (Real-World South Africa): According to statistics, the probability of being involved in a car accident in South Africa is relatively high compared to other countries. This is related to factors like road conditions, driver behavior, and vehicle maintenance. Understanding this risk is crucial for taking precautions such as driving defensively, maintaining your vehicle, and avoiding driving under the influence. Insurance premiums reflect this higher risk. Guided Practice (With Solutions)

Question 1: A spinner has 4 equal sections colored red, blue, green, and yellow. You spin the spinner twice. What is the probability of landing on red both times?

Solution: P(Red on first spin) = 1/4 P(Red on second spin) = 1/4 Since the spins are independent: P(Red and Red) = (1/4) (1/4) = 1/16 Therefore, the probability of landing on red both times is 1/

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6. Question 2: A bag contains 6 apples and 4 oranges. You randomly select two fruits without replacement. What is the probability of selecting an apple first and then an orange?

Solution: P(Apple first) = 6/10 = 3/5 After selecting an apple, there are now 5 apples and 4 oranges, making a total of 9 fruits. P(Orange second, given apple first) = 4/9 P(Apple then Orange) = (3/5) (4/9) = 12/45 = 4/15 The probability of selecting an apple first and then an orange is 4/

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5. Question 3: A survey shows that 70% of people in a town support a new development project.