Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Define and calculate the probability of a simple event.
• Identify certain, impossible, likely, and unlikely events.
• Use probability to predict outcomes.
• Understand and apply relative frequency.
• Calculate the probability of combined events.

Lesson summary

Probability is all about understanding the chances of something happening. In our daily lives in South Africa, we constantly encounter situations where we assess risks and possibilities, even if we don't realize it. For instance, deciding whether to take an umbrella based on the weather forecast, understanding the odds of winning the Lotto, or evaluating the potential risks and benefits of different career paths all involve probability. This week, we'll explore the basic concepts of probability and learn how to calculate the likelihood of simple events.

Lesson notes

2.1 Basic Definitions: Experiment: An activity involving chance that leads to results called outcomes.

Example:* Rolling a die.

Outcome: A possible result of an experiment.

Example:* Rolling a '3' on a die.

Sample Space: The set of all possible outcomes of an experiment.

Example:* For rolling a die, the sample space is {1, 2, 3, 4, 5, 6}.

Event: A specific outcome or set of outcomes that we are interested in.

Example:* Rolling an even number on a die (the event is {2, 4, 6}).

Probability: A measure of how likely an event is to occur. It is expressed as a number between 0 and 1 (or as a percentage between 0% and 100%). 2.2 Calculating Probability: The probability of an event (E) is calculated as: P(E) = (Number of favorable outcomes) / (Total number of possible outcomes) Important

Note: This formula assumes that all outcomes in the sample space are equally likely. 2.3 Expressing Probability: Probability can be expressed in three ways: Fraction:

Example:* 1/2 Decimal:

Example:* 0.5 Percentage:

Example:* 50% All three representations are equivalent. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100. 2.4 Types of Events: Certain Event: An event that will definitely happen. Its probability is 1 (or 100%).

Example:* The sun will rise tomorrow.

Impossible Event: An event that cannot happen. Its probability is 0 (or 0%).

Example:* Rolling a '7' on a standard six-sided die.

Likely Event: An event that has a high probability of happening (closer to 1 or 100%).

Unlikely Event: An event that has a low probability of happening (closer to 0 or 0%). 2.5 Relative Frequency: Sometimes, we cannot determine the exact probability of an event theoretically. In such cases, we can estimate the probability based on observed data using the concept of relative frequency. Relative Frequency = (Number of times the event occurred) / (Total number of trials)

Example: If a coin is tossed 100 times and lands on heads 55 times, the relative frequency of heads is 55/100 = 0.55 or 55%. This is an estimate of the probability of getting heads. 2.6 Combined Events: Sometimes, we want to find the probability of two or more events happening. For simple independent events (where one event doesn't affect the other), we can multiply their individual probabilities. P(A and B) = P(A) * P(B)

Example: What is the probability of rolling a '6' on a die and flipping heads on a coin? P(rolling a '6') = 1/6 P(flipping heads) = 1/2 P(rolling a '6' AND flipping heads) = (1/6) (1/2) = 1/12 2.7 Worked

Examples: Example 1: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of picking a blue marble at random? Express your answer as a fraction, decimal, and percentage.

Solution: Total number of marbles: 5 + 3 + 2 = 10 Number of blue marbles: 3 P(blue marble) = 3/10 Decimal: 3/10 = 0.3 Percentage: 0.3 100 = 30% Answer: The probability of picking a blue marble is 3/10, 0.3, or 30%.

Example 2: A spinner has 4 equal sections colored red, blue, green, and yellow. What is the probability of the spinner landing on red or blue?

Solution: Total number of sections: 4 Number of sections that are red or blue: 2 P(red or blue) = 2/4 = 1/2 Decimal: 1/2 = 0.5 Percentage: 0.5 100 = 50% Answer: The probability of the spinner landing on red or blue is 1/2, 0.5, or 50%.

Example 3: A survey of 500 South African households found that 350 have a television. Based on this data, what is the estimated probability that a randomly selected South African household has a television?

Solution: Total number of households surveyed: 500 Number of households with a television: 350 Relative frequency (estimated probability) = 350/500 = 7/10 Decimal: 7/10 = 0.7 Percentage: 0.7 100 = 70% Answer: The estimated probability is 7/10, 0.7, or 70%. Guided Practice (With Solutions)

Question 1: A standard six-sided die is rolled. What is the probability of rolling a number greater than 4?

Solution: Sample Space: {1, 2, 3, 4, 5, 6} Favorable Outcomes (numbers greater than 4): {5, 6} Number of favorable outcomes: 2 Total number of possible outcomes: 6 P(rolling a number greater than 4) = 2/6 = 1/3 Answer: 1/

3. Explanation: We identified the favorable outcomes within the sample space and applied the basic probability formula.* Question 2: A bag contains 8 apples and 12 oranges. If you pick a fruit at random, what is the probability of picking an orange? Express your answer as a percentage.

Solution: Total number of fruits: 8 + 12 = 20 Number of oranges: 12 P(picking an orange) = 12/20 = 3/5 Percentage: (3/5) 100 = 60% Answer: 60%.

Explanation: We calculated the total number of fruits, then found the probability of picking an orange and converted it to a percentage.* Question 3: What is the probability of flipping a coin and getting tails, and then rolling a die and getting a 3?