Lesson Notes By Weeks and Term v5 - Grade 11
Probability: predicting outcomes and risk – Week 10 focus
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Probability: predicting outcomes and risk – Week 10 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Mathematical Literacy
Class: Grade 11
Term: Term 4
Week: 10
Theme: General lesson support
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Probability is a crucial concept in Mathematical Literacy that helps us understand and predict the likelihood of events happening. It’s not just about tossing coins or rolling dice; it's about making informed decisions in everyday life, from managing personal finances to understanding health risks and evaluating insurance policies. In South Africa, where many face socio-economic challenges and uncertainties, understanding probability is vital for making responsible choices and planning for the future. Being able to assess risk, predict potential outcomes, and understand the odds helps to navigate complex situations and empowers individuals to take control of their lives.
2.1 Basic Probability: Probability is the measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means the event is impossible, and 1 means the event is certain. It can also be expressed as a percentage (0% to 100%).
Formula: Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes)
Example 1: A standard six-sided die is rolled. What is the probability of rolling a 4? There is only one favorable outcome (rolling a 4). There are six possible outcomes (1, 2, 3, 4, 5, 6). Probability (rolling a 4) = 1/6 = 0.1667 (approximately) = 16.67% Example 2: A bag contains 5 red marbles and 3 blue marbles. What is the probability of picking a red marble at random? Number of favorable outcomes (red marbles) = 5 Total number of marbles = 5 + 3 = 8 Probability (picking a red marble) = 5/8 = 0.625 = 62.5% 2.2 Sample Space and Events: Sample Space: The set of all possible outcomes of an experiment. For example, the sample space for rolling a die is {1, 2, 3, 4, 5, 6}. The sample space for flipping a coin is {Heads, Tails}.
Event: A specific outcome or set of outcomes from the sample space. For example, "rolling an even number" on a die is an event represented by {2, 4, 6}. 2.3 Compound Events: Compound events involve two or more events happening together or in sequence.
Independent Events: Events where the outcome of one event does NOT affect the outcome of the other. For example, flipping a coin twice. The result of the first flip does not influence the result of the second flip. The probability of two independent events A and B happening is: P(A and B) = P(A) P(B)
Example: What is the probability of flipping a coin and getting heads, AND then rolling a die and getting a 3? P(Heads) = 1/2 P(Rolling a 3) = 1/6 P(Heads and Rolling a 3) = (1/2) (1/6) = 1/12 = 0.0833 = 8.33% Dependent Events: Events where the outcome of one event DOES affect the outcome of the other. For example, drawing two cards from a deck without replacing the first card. The probability of the second card depends on what card was drawn first.
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: A bag contains 5 red balls and 3 blue balls. You pick one ball at random, and then pick another without replacing the first. What is the probability of picking a red ball, and then another red ball? P(Red on first pick) = 5/8 If you pick a red ball on the first pick, there are now 4 red balls and 3 blue balls left, for a total of 7 balls. P(Red on second pick, given red on first pick) = 4/7 P(Red and Red) = (5/8) (4/7) = 20/56 = 5/14 = 0.3571 = 35.71% 2.4 Predicting Outcomes and Risk Assessment: Probability is used to predict outcomes and assess risk. A higher probability of an undesirable event indicates a higher risk.
Example: Loan Default Risk A bank assesses the probability of a person defaulting on a loan based on their credit score, income, and employment history. If the probability of default is high (e.g., 20%), the bank might charge a higher interest rate or require collateral to mitigate the risk.
Example: Investing in Stocks The probability of a stock increasing in value can be estimated based on market trends and company performance. A higher probability of increase might encourage investment, but there's always a risk of loss. 2.5 Impact of Sample Size: A larger sample size generally leads to more reliable probability predictions. A small sample size can be misleading and not representative of the overall population.
Example: You flip a coin 10 times and get 7 heads. This suggests a probability of 70% for heads.
However, if you flip the coin 1000 times and get 510 heads, the probability is closer to 51%, which is a better estimate because it is based on more data. Guided Practice (With Solutions)
Question 1: A survey of 100 Grade 11 learners showed that 60 like soccer, 30 like rugby, and 10 like both. What is the probability that a randomly selected learner likes: (a) Soccer? (b) Rugby? (c) Soccer or Rugby?
Solution: (a) P(Soccer) = (Number of learners who like soccer) / (Total number of learners) = 60/100 = 3/5 = 0.6 = 60% (b) P(Rugby) = (Number of learners who like rugby) / (Total number of learners) = 30/100 = 3/10 = 0.3 = 30% (c) To find the probability of Soccer OR Rugby, we need to avoid counting the learners who like both twice. So: P(Soccer or Rugby) = P(Soccer) + P(Rugby) - P(Soccer and Rugby) = (60/100) + (30/100) - (10/100) = 80/100 = 4/5 = 0.8 = 80% Question 2: A security company estimates that the probability of a house being burgled in a certain suburb is 0.
0
5. If there are 500 houses in the suburb, how many burglaries are expected to occur in a year?
Solution: Expected number of burglaries = Probability of burglary Number of houses = 0.05 500 = 25 burglaries. This is a prediction based on probability; the actual number may be higher or lower.