Lesson Notes By Weeks and Term v5 - Grade 10
Probability – Week 5 focus
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Probability – Week 5 focus
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Subject: Mathematics
Class: Grade 10
Term: Term 4
Week: 5
Theme: General lesson support
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Probability is the branch of mathematics that deals with the likelihood of an event occurring. It provides a framework for understanding and quantifying uncertainty. Understanding probability is crucial in everyday life. For South African learners, it's relevant to understanding things like the chances of winning the Lotto, interpreting weather forecasts (probability of rain), understanding risk in sports (injury probability), and making informed decisions about insurance or investments later in life. In a country facing diverse challenges and opportunities, probability helps us assess risks and make predictions based on available data.
Theoretical Probability: Theoretical probability is based on reasoning about the nature of the event. It is the ratio of the number of favorable outcomes to the total number of possible outcomes, assuming all outcomes are equally likely.
Formula: P(Event) = (Number of favorable outcomes) / (Total number of possible outcomes)
Example 1: A fair 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).
Therefore, P(rolling a 4) = 1/6 Example 2: A bag contains 3 red balls, 2 blue balls, and 5 green balls. What is the probability of randomly selecting a red ball? Number of favorable outcomes (red balls) = 3 Total number of possible outcomes (total balls) = 3 + 2 + 5 = 10 Therefore, P(selecting a red ball) = 3/10 Experimental Probability: Experimental probability is based on the results of an experiment or a series of trials. It is the ratio of the number of times an event occurs to the total number of trials.
Formula: P(Event) = (Number of times the event occurs) / (Total number of trials)
Example 3: A coin is flipped 50 times. Heads appear 28 times and tails appear 22 times. What is the experimental probability of getting heads? Number of times heads occurs = 28 Total number of trials = 50 Therefore, P(heads) = 28/50 = 14/25 Example 4: A survey of 100 learners at a school near Durban found that 60 preferred soccer, 30 preferred rugby and 10 prefer other sports. What is the experimental probability that a student, selected at random, will prefer soccer? Number of learners preferring soccer = 60 Total number of learners surveyed = 100 Therefore, P(soccer) = 60/100 = 3/5 Theoretical vs.
Experimental Probability: Theoretical probability represents what should happen in an ideal situation. Experimental probability represents what actually happens when an experiment is performed. As the number of trials increases, the experimental probability tends to get closer to the theoretical probability (this is known as the Law of Large Numbers).
Mutually Exclusive Events: Two events are mutually exclusive (or disjoint) if they cannot occur at the same time. If A and B are mutually exclusive events, then P(A and B) =
0. The probability of either A or B occurring is the sum of their individual probabilities.
Formula: P(A or B) = P(A) + P(B) (if A and B are mutually exclusive)
Example 5: Rolling a die. The events "rolling a 2" and "rolling a 5" are mutually exclusive because you cannot roll both a 2 and a 5 at the same time. If P(rolling a 2) = 1/6 and P(rolling a 5) = 1/6, then P(rolling a 2 or a 5) = 1/6 + 1/6 = 2/6 = 1/3 Example 6: In a class, a learner cannot be both in Grade 10A and Grade 10B simultaneously. If P(learner in 10A) = 0.4 and P(learner in 10B) = 0.3, then P(learner in 10A or 10B) = 0.4 + 0.3 = 0.7 Venn Diagrams and Probability: Venn diagrams are useful for visualizing and calculating probabilities, especially when dealing with events that are not mutually exclusive (meaning they can occur simultaneously).
Example 7: In a class of 30 learners, 18 play soccer, 12 play rugby, and 5 play both soccer and rugby. What is the probability that a randomly selected learner plays soccer or rugby? Draw a Venn diagram with two overlapping circles, one for soccer (S) and one for rugby (R). The intersection (S ∩ R) represents learners who play both, so write 5 in the intersection.
Learners who play only soccer: 18 - 5 = 13 Learners who play only rugby: 12 - 5 = 7 Total learners playing soccer or rugby: 13 + 5 + 7 = 25 P(soccer or rugby) = 25/30 = 5/6 Guided Practice (With Solutions)
Question 1: A spinner has 8 equally sized sections, numbered 1 to
8. What is the theoretical probability of landing on an even number?
Solution: Favorable outcomes (even numbers): 2, 4, 6, 8 (4 outcomes)
Total possible outcomes: 8 P(even number) = 4/8 = 1/2
Commentary: This is a straightforward application of the theoretical probability formula. We identified the favorable outcomes and divided by the total outcomes.
Question 2: A student conducted an experiment by rolling a die 60 times. They recorded the number 3 appearing 12 times. What is the experimental probability of rolling a 3?
Solution: Number of times 3 appeared: 12 Total number of trials: 60 P(rolling a 3) = 12/60 = 1/5
Commentary: This question focuses on calculating experimental probability, emphasizing that it's based on observed data.
Question 3: A box contains 5 apples and 7 oranges. If you randomly select one fruit, what is the probability of picking an apple OR an orange?
Solution: These are mutually exclusive events because you can only pick one fruit at a time. P(apple) = 5 / (5+7) = 5/12 P(orange) = 7 / (5+7) = 7/12 P(apple or orange) = P(apple) + P(orange) = 5/12 + 7/12 = 12/12 = 1
Commentary: This problem highlights the concept of mutually exclusive events and the fact that picking either an apple or orange covers the whole sample space, so the probability must be 1.