Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate probability of single and combined events.
• Interpret probabilities to assess risks.
• Use probability to predict the likelihood of events.
• Distinguish between independent and dependent events.
• Understand experimental vs. theoretical probability.

Lesson summary

Probability is a fundamental concept in Mathematical Literacy that allows us to understand and quantify the likelihood of events occurring. It's not just about flipping coins or rolling dice; it's a powerful tool for making informed decisions in everyday life. In South Africa, understanding probability helps us navigate various situations, from understanding the risks associated with investments to interpreting weather forecasts and making informed decisions about insurance. It is particularly relevant when considering issues such as the lottery (Lotto), and the statistical likelihood of winning, and risks associated with various social and health issues like HIV/AIDS transmission.

Lesson notes

2.1 Basic Probability: Probability is a 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 (denoted as P(event)) is calculated as: P(event) = (Number of favourable outcomes) / (Total number of possible outcomes)

Example 1: A bag contains 5 red marbles and 3 blue marbles. What is the probability of randomly picking a red marble?

Solution: Number of favourable outcomes (red marbles) = 5 Total number of possible outcomes (total marbles) = 5 + 3 = 8 P(red marble) = 5/8 = 0.625 This means there is a 62.5% chance of picking a red marble. 2.2 Experimental vs.

Theoretical Probability: Theoretical Probability: This is what we expect to happen based on mathematical calculations, assuming a fair and unbiased situation. It is calculated as shown above.

Experimental Probability: This is what actually happens when we conduct an experiment (e.g., flipping a coin many times).

It is calculated as: P(event) = (Number of times the event occurs) / (Total number of trials)

Example 2: A coin is flipped 20 times. Heads comes up 12 times.

Theoretical probability of getting heads: 1/2 = 0.5 Experimental probability of getting heads: 12/20 = 0.6 The experimental probability may not always match the theoretical probability, especially with a small number of trials.

However, as the number of trials increases, the experimental probability tends to get closer to the theoretical probability (Law of Large Numbers). 2.3 Independent and Dependent Events: Independent Events: Two events are independent if the outcome of one event does not affect the outcome of the other event. The probability of both events A and B occurring is: P(A and B) = P(A) * P(B)

Example 3: You roll a die and then flip a coin. What is the probability of rolling a 4 and getting heads? P(rolling a 4) = 1/6 P(getting heads) = 1/2 P(rolling a 4 and getting heads) = (1/6) (1/2) = 1/12 Dependent Events: Two events are dependent if the outcome of one event does affect the outcome of the other event. This often involves situations where an item is removed and not replaced. The probability of both events A and B occurring is: 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 4: A bag contains 5 red balls and 3 blue balls. You pick one ball, do not replace it, and then pick another ball. What is the probability of picking a red ball followed by another red ball? P(first ball is red) = 5/8 P(second ball is red, given the first was red) = 4/7 (because there are now only 4 red balls and 7 total balls left) P(red then red) = (5/8) (4/7) = 20/56 = 5/14 2.4 Combined Events (OR): When calculating the probability of event A OR event B happening, we need to consider if the events are mutually exclusive (cannot happen at the same time) or not.

Mutually Exclusive Events: P(A or B) = P(A) + P(B)

Example 5: What is the probability of rolling a 2 or a 5 on a single die? P(rolling a 2) = 1/6 P(rolling a 5) = 1/6 P(rolling a 2 or a 5) = (1/6) + (1/6) = 2/6 = 1/3 Non-Mutually Exclusive Events: P(A or B) = P(A) + P(B) - P(A and B)

Example 6: What is the probability of drawing a heart or a king from a standard deck of cards? P(drawing a heart) = 13/52 = 1/4 P(drawing a king) = 4/52 = 1/13 P(drawing a heart AND a king) = 1/52 (the king of hearts) P(drawing a heart OR a king) = (1/4) + (1/13) - (1/52) = 16/52 = 4/13 2.5 Predicting Outcomes and Assessing Risk: Probability can be used to predict the likelihood of future events. This is particularly important in risk assessment. For example, insurance companies use probability to determine premiums based on the likelihood of accidents or other insurable events. Understanding probability allows individuals to make more informed decisions about their health, finances, and safety.

Example 7: Lotto The South African Lotto requires you to pick 6 numbers from 1 to

5

2. What is the probability of winning the jackpot if you buy one ticket? The probability of winning is 1 in 20,358,

5

2

0. This is a very, very small probability, which illustrates the high risk associated with playing the lotto. Guided Practice (With Solutions)

Question 1: A weather forecast predicts a 30% chance of rain tomorrow. What is the probability that it will not rain tomorrow?

Solution: The probability of an event NOT happening is 1 - P(event). P(rain) = 30% = 0.3 P(no rain) = 1 - 0.3 = 0.7 Therefore, there is a 70% chance that it will not rain tomorrow.

Question 2: A hospital has 120 patients. Statistically, 5% of the patients are expected to have a specific rare disease. How many patients are expected to have the disease?

Solution: Probability of having the disease = 5% = 0.05 Total number of patients = 120 Expected number of patients with the disease = 0.05 120 = 6 Therefore, 6 patients are expected to have the disease.

Question 3: A box contains 8 green apples and 4 red apples.