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 estimate risk.
• Differentiate between theoretical and experimental probability.
• Understand and apply 'independent' and 'dependent' events.
• Critically evaluate the use of probability in real-life.

Lesson summary

This week, we delve into the fascinating world of probability, specifically focusing on predicting outcomes and assessing risk. Probability is not just about flipping coins or rolling dice; it's a fundamental tool for making informed decisions in our daily lives, especially in a diverse and sometimes unpredictable South Africa. From understanding the chances of winning the Lotto to evaluating the risks associated with different investments or health choices, probability equips us with the skills to navigate the world with greater confidence. This topic is crucial for developing critical thinking and informed citizenship.

Lesson notes

2.1 What is Probability? Probability is a measure of how likely an event is to occur. It's expressed as a number between 0 and 1, where 0 means the event is impossible, and 1 means the event is certain. Probability can also be expressed as a percentage (multiplying the decimal by 100).

Formula: Probability of an Event = (Number of favourable outcomes) / (Total number of possible outcomes) 2.2 Sample Space and Events Sample Space: The set of all possible outcomes of a random experiment. For example, when rolling a standard six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.

Event: A subset of the sample space. For example, rolling an even number on a die is an event, and its possible outcomes are {2, 4, 6}. 2.3 Theoretical vs. Experimental Probability Theoretical Probability: The probability of an event based on mathematical calculations, assuming all outcomes are equally likely. This is what we calculate using the formula above.

Experimental Probability: The probability of an event based on observations from an experiment or real-world data.

It's calculated as: (Number of times the event occurs) / (Total number of trials).

Example: Tossing a fair coin. Theoretically, the probability of getting heads is 1/

2. However, if you toss the coin 10 times and get 7 heads, the experimental probability of getting heads in that experiment is 7/

1

0. The difference between theoretical and experimental probability often arises due to random variation in real-world experiments. The more trials you conduct, the closer the experimental probability is likely to get to the theoretical probability (Law of Large Numbers). 2.4 Independent vs.

Dependent Events Independent Events: Events where 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: P(A and B) = P(A) P(B).

Example: Flipping a coin and rolling a die. The result of the coin flip doesn't affect the result of the die roll.

Dependent Events: Events where the outcome of one event does affect the outcome of the other. 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) is the probability of B occurring given that A has already occurred.

Example: Drawing two cards from a deck without replacement. The probability of drawing a second card of a particular suit depends on what card was drawn first. 2.5 Compound Events These are events that combine two or more simpler events. "OR" Rule: P(A or B) = P(A) + P(B) - P(A and B). We subtract P(A and B) to avoid double-counting when A and B are not mutually exclusive (i.e., they can both happen).

Example: What is the probability of drawing a king OR a heart from a standard deck of cards? P(King) = 4/52, P(Heart) = 13/52, P(King and Heart) = 1/

5

2. So, P(King or Heart) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13. 2.6 Predicting Outcomes and Assessing Risk Probability is essential for predicting outcomes and assessing risk. By understanding the likelihood of different events, we can make informed decisions.

Example: A company in South Africa wants to launch a new product. They conduct market research and estimate that there is a 60% chance the product will be successful. This probability helps them assess the risk of investing in the new product. A higher probability of success implies lower risk, but it's still important to consider the potential costs and benefits.

Worked example

Example 1: The Lotto

The South African Lotto requires you to choose 6 numbers from 1 to

5

2. What is the probability of winning the jackpot? (This is a very complex calculation, but demonstrates the principles)

Solution:

The number of possible combinations is calculated using combinations formula: ⁵²C₆ = 52! / (6! * (52-6)!) = 20,358,

5

2

0.

The probability of winning the jackpot is therefore 1 / 20,358,520, a very small number! This highlights the low probability and high risk involved in playing the Lotto.