Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate probabilities of single and combined events.
• Use probability to predict outcomes.
• Analyse probabilities to assess risks in real-life scenarios.
• Compare theoretical and experimental probability.
• Understand independent and dependent events.

Lesson summary

This week, we delve into the fascinating world of probability, focusing on predicting outcomes and understanding risk. Probability is a vital tool in Mathematical Literacy because it helps us make informed decisions in various aspects of life, from understanding the chances of winning the lotto (and therefore whether it's a wise use of our money) to assessing the risks associated with driving in different conditions. In a South African context, understanding probability can empower learners to make sound financial choices, evaluate health risks, and participate more effectively in democratic processes where understanding polls and statistics is crucial.

Lesson notes

What is Probability? Probability is a 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. Probability can also be expressed as a percentage (0% to 100%).

Theoretical Probability: Theoretical probability is based on reasoning and calculation, assuming all outcomes are equally likely.

It's calculated as: P(event) = (Number of favourable outcomes) / (Total number of possible outcomes)

Example 1: Flipping a Fair Coin What is the probability of getting heads when flipping a fair coin?

Total possible outcomes: 2 (Heads or Tails)

Favourable outcome: 1 (Heads) P(Heads) = 1/2 = 0.5 = 50% Example 2: Rolling a Fair Die What is the probability of rolling a 4 on a fair six-sided die?

Total possible outcomes: 6 (1, 2, 3, 4, 5, 6)

Favourable outcome: 1 (4) P(Rolling a 4) = 1/6 ≈ 0.167 ≈ 16.7% Experimental Probability: Experimental probability is based on the results of an experiment or observation.

It's calculated as: P(event) = (Number of times the event occurred) / (Total number of trials)

Example 3: Flipping a Coin Multiple Times You flip a coin 20 times and get heads 12 times. What is the experimental probability of getting heads?

Total number of trials: 20 Number of times heads occurred: 12 P(Heads) = 12/20 = 0.6 = 60% Important

Note: Experimental probability may differ from theoretical probability, especially with a small number of trials. The more trials you conduct, the closer the experimental probability is likely to get to the theoretical probability (Law of Large Numbers).

Independent Events: Independent events are events where the outcome of one event does not affect the outcome of the other. P(A and B) = P(A) P(B)* Example 4: Independent Events - Two Coin Flips What is the probability of getting heads on both of two consecutive coin flips? Assume the coin is fair. P(Heads on first flip) = 1/2 P(Heads on second flip) = 1/2 P(Heads on both flips) = (1/2) (1/2) = 1/4 = 0.25 = 25%* Dependent Events: Dependent events are events where the outcome of one event does affect the outcome of the other. The probability of the second event happening depends on whether the first event happened. P(A and B) = P(A) P(B|A)*, where P(B|A) is the probability of B given that A has already occurred.

Example 5: Dependent Events - Drawing Cards Without Replacement A standard deck of cards has 52 cards. You draw a card and don't put it back in the deck. What is the probability of drawing an Ace first, and then drawing a King? P(Drawing an Ace first) = 4/52 = 1/13 (since there are 4 Aces) P(Drawing a King second given that you drew an Ace first) = 4/51 (since there are still 4 Kings, but only 51 cards left) P(Ace then King) = (1/13) (4/51) = 4/663 ≈ 0.006 ≈ 0.6%* Predicting Outcomes: We can use probability to predict how many times an event is likely to occur in a series of trials. Expected Number of Occurrences = P(event) (Number of trials)* Example 6: Predicting Die Rolls You roll a fair six-sided die 60 times. How many times would you expect to roll a 5? P(Rolling a 5) = 1/6 Number of trials = 60 Expected Number of 5s = (1/6) 60 = 10 Therefore, you would expect to roll a 5 approximately 10 times.

Understanding Risk: Risk is the possibility of loss or harm. We can use probability to assess and manage risk. Higher probabilities of undesirable events indicate higher risk.

Example 7: Insurance An insurance company calculates the probability of a car accident for different age groups. If the probability of an accident for a driver aged 18-25 is 0.15 (15%), and for a driver aged 40-50 is 0.05 (5%), the insurance company will likely charge higher premiums for the younger driver because the risk of an accident is higher.

Tree Diagrams: Tree diagrams are helpful for visualizing and calculating probabilities of events that occur in sequence. Each branch represents a possible outcome, and the probabilities are written along the branches. To find the probability of a sequence of events, multiply the probabilities along the corresponding branches.

Example 8: Drawing Balls from a Bag (Without Replacement) A bag contains 3 red balls and 2 blue balls. You draw two balls without replacement. What is the probability of drawing two red balls?

First Draw: P(Red) = 3/5 P(Blue) = 2/5 Second Draw (given the first ball was red): P(Red) = 2/4 (since one red ball has been removed) P(Blue) = 2/4 Second Draw (given the first ball was blue): P(Red) = 3/4 P(Blue) = 1/4 (since one blue ball has been removed) To find the probability of two red balls, we follow the "Red" branch on the first draw and then the "Red" branch on the second draw, given that the first ball was red. P(Red, then Red) = (3/5) * (2/4) = 6/20 = 3/10 = 0.3 = 30% Guided Practice (With Solutions)

Question 1: A bag contains 5 green marbles and 3 yellow marbles. If you randomly select one marble, what is the probability that it is green?