D2. Probability
Specific Expectations
Probability
D2.1
describe the difference between independent and dependent events, and explain how their probabilities differ, providing examples
- independent events:
- tossing a coin, then rolling a die
- dependent events:
- a card is drawn at random from a standard deck of playing cards, and without replacing it, a second card is chosen
- Two events are independent when the outcome of one event does not affect the outcome of the other event.
- Two events are dependent when the outcome of the first event affects the outcome of the second event.
Note
- The probabilities for independent and dependent events can be compared when based on the same event with slightly different conditions. For example, the probability of selecting two names from a bag with replacement versus the probability of selecting two names from a bag without replacement.
Provide students with statements about a variety of different probability experiments and have them identify and provide a rationale for whether the experiments are dealing with independent or dependent events. Statements might include the following:
- You have five cubes. Three are red and two are blue. Create a scenario that involves independent events, and another scenario that involves dependent events. Explain how the probabilities differ.
- You have twenty-five numbers. Create a scenario that involves independent events, and another scenario that involves dependent events. Explain how the probabilities differ.
- You have thirty names on a class list. Create a scenario that involves independent events, and another scenario that involves dependent events. Explain how the probabilities differ.
- You have one die and two coins. Create a scenario that involves independent events, and another scenario that involves dependent events. Explain how the probabilities differ.
Provide students with the following two tree diagrams, and have them identify how they are similar and how they are different.
Have students identify which tree diagram illustrates drawing two marbles independently from each other, and which one illustrates drawing two marbles where the outcome of the second draw depends on the first.
Ask students to determine the probability of drawing two red marbles in the independent event scenario. Ask how this differs from the probability of drawing two red marbles in the dependent event scenario.
D2.2
determine and compare the theoretical and experimental probabilities of two independent events happening and of two dependent events happening
- probability experiments involving independent events:
- roll two or more dice, repeat 20 times
- toss two or more coins, repeat 15 times
- roll one or more dice and toss one or more coins, repeat 30 times
- pick an item from a bag two or more times, replace after each pick, repeat 25 times
- probability experiments involving dependent events:
- pick an item from a bag two or more times, do not replace after each pick, repeat 25 times
- randomly pick two or more names from a class list
- The more trials completed in an experiment, the closer the experimental probability will be to the theoretical probability.
- The sum of the probability of all possible outcomes is 1 or 100%.
- The probability of an event can be used to predict the likelihood of that event happening again in the future.
- Tree diagrams are helpful to determine all the possible outcomes for two independent events and two dependent events.
Note
- “Odds in favour” is a comparison of the probability that an event will occur with the probability that the event will not occur (complementary events). For example, the probability that the sum of two dice is 2 is $$\frac{1}{36}$$ and the probability that the sum of two dice is not 2 is $$\frac{35}{36}$$. The odds in favour of rolling a sum of 2 is $$\frac{1}{36}$$ : $$\frac{35}{36}$$ or 1:35, since the fractions are both relative to the same whole
Have students design two probability simulations in which two numbers from 1 to 10 are randomly generated for 10, 30, and 100 trials. One of the simulations should allow for numbers to be repeated (independent events). The other simulation should not allow for numbers to be repeated (dependent events).
Have students determine the experimental probabilities of only even numbers being generated by each of the simulations they designed in Sample Task 1 and compare these to the theoretical probabilities for each simulation.
