Lesson Notes By Weeks and Term v5 - Grade 11
Probability – Week 10 focus
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Probability – Week 10 focus
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Subject: Mathematics
Class: Grade 11
Term: 3rd Term
Week: 10
Theme: General lesson support
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Probability is the bedrock of understanding chance and uncertainty. It is not just a mathematical concept confined to textbooks; it permeates our daily lives, influencing decisions from whether to carry an umbrella based on weather forecasts to understanding the risks associated with investments or the likelihood of winning a lottery. In the South African context, understanding probability is crucial for informed decision-making in areas such as understanding insurance policies (e.g., car insurance, medical aid), assessing the risks associated with entrepreneurship, and even interpreting statistical data related to public health or crime rates.
2.1 Independent Events: Two events are considered independent if the outcome of one event does not affect the probability of the other event occurring. If event A and event B are independent, then: P(A and B) = P(A) * P(B)
Example: Tossing a fair coin twice. The outcome of the first toss does not influence the outcome of the second toss. The probability of getting heads on the first toss is 1/2, and the probability of getting heads on the second toss is also 1/
2. Therefore, the probability of getting heads on both tosses is (1/2) * (1/2) = 1/
4. Example (South African context): A factory producing light bulbs finds that 2% of bulbs are defective. Two bulbs are selected at random. What is the probability that both bulbs are defective, assuming the selection of one bulb doesn't affect the probability of the other being defective (i.e., large production)? P(Bulb 1 defective) = 0.02 P(Bulb 2 defective) = 0.02 P(Both defective) = 0.02 * 0.02 = 0.0004 or 0.04% 2.2 Dependent Events: Two events are considered dependent if the outcome of one event does affect the probability of the other event occurring. If event A and event B are dependent, then: P(A and B) = P(A) * P(B|A) Where P(B|A) represents the conditional probability of event B occurring given that event A has already occurred.
Example: Drawing two cards from a deck without replacement. The outcome of the first draw influences the probability of the second draw because the number of cards in the deck has changed.
Example (South African context): A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. What is the probability that the first ball is red and the second ball is blue? P(First ball is red) = 5/8 P(Second ball is blue | First ball is red) = 3/7 (since there are now only 7 balls left, 3 of which are blue) P(First red and second blue) = (5/8) * (3/7) = 15/56 2.3 Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time. If event A and event B are mutually exclusive, then: P(A or B) = P(A) + P(B)
Example: Tossing a coin. The result is either heads or tails, but not both simultaneously.
Example (South African context): Consider a raffle with 100 tickets. You buy 5 tickets. One ticket will be drawn to win a prize. What is the probability that you either win the first prize OR win the second prize if the second prize is drawn from the remaining tickets after the first prize? Assume only one prize per ticket. P(Winning first prize) = 5/100 = 1/20 P(Winning second prize) = 0 (If you win the first prize, you can't win the second). If you don't win the first prize (probability 95/100), then the probability of winning the second prize with one of your tickets is 5/99 P(Winning first or second prize) = P(Winning first) + P(Not winning first) P(Winning second | Not winning first) = 1/20 + (95/100) (5/99) = 1/20 + (19/20) * (5/99) = 1/20 + 19/396 = (99 + 95)/1980 = 194/1980 = 97/990 2.4 Non-Mutually Exclusive Events: Two events are non-mutually exclusive if they can occur at the same time. If event A and event B are non-mutually exclusive, then: P(A or B) = P(A) + P(B) - P(A and B) The subtraction of P(A and B) is to avoid double-counting the outcomes that belong to both events.
Example: Drawing a card from a standard deck. What is the probability of drawing a heart or a king? You can draw a card that is both a heart and a king (the King of Hearts).
Example (South African context): In a class of 30 learners, 15 take mathematics, 10 take physical science, and 5 take both. What is the probability that a randomly selected learner takes mathematics or physical science? P(Mathematics) = 15/30 = 1/2 P(Physical Science) = 10/30 = 1/3 P(Mathematics and Physical Science) = 5/30 = 1/6 P(Mathematics or Physical Science) = (1/2) + (1/3) - (1/6) = (3 + 2 - 1)/6 = 4/6 = 2/3 2.5 Tree Diagrams: Tree diagrams are visual tools used to represent the probabilities of different outcomes in a sequence of events. Each branch of the tree represents a possible outcome, and the probabilities are written along the branches. To find the probability of a particular sequence of outcomes, you multiply the probabilities along the corresponding branches.
Example: A coin is tossed twice. Draw a tree diagram and find the probability of getting heads on both tosses. ``` / H (1/2) -> H (1/2) = HH (1/4) / Start --- \ T (1/2) -> H (1/2) = TH (1/4) \ / H (1/2) -> T (1/2) = HT (1/4) / Start --- \ T (1/2) -> T (1/2) = TT (1/4) \ ``` P(HH) = (1/2) * (1/2) = 1/4 Example (South African context): A soccer team has a 60% chance of winning a game if it rains and a 80% chance of winning if it doesn't rain. The weather forecast predicts a 40% chance of rain. What is the probability that the team will win?