## D2. Probability

### Specific Expectations

#### Probability

D2.1

solve various problems that involve probability, using appropriate tools and strategies, including Venn and tree diagrams

**Venn diagram:**

- Venn diagrams can be used to understand the relationship of given probabilities involving multiple events in order to solve a problem. The sum of the components of a Venn diagram is 100% of the total population or sample being referenced.
- Tree diagrams can be used to determine all possible combinations of outcomes for two or more events that are either independent or dependent.

*Note*

- Sample space diagrams are another visual way of recording all possible outcomes for two events. The diagram below shows the possibilities when two coins are tossed.

Give students data that they can display in a Venn diagram and then use to determine the probability of an event. For example, in a class of 30 students, 19 students like mystery novels, 17 students like adventure novels, and 15 students like both types of novels. What is the probability of students liking neither adventure nor mystery novels?

D2.2

determine and compare the theoretical and experimental probabilities of multiple independent events happening and of multiple dependent events happening

**independent events:**- rolling dice simultaneously
- tossing coins
- drawing an item from a bag and returning the item to the bag after each draw

**probability experiments involving dependent events:**- drawing an item from a bag and not returning the item to the bag before drawing again
- randomly picking a name from a class list when the name can only be selected once

- Two events are independent if the probability of one does not affect the probability of the other. For example, the probability for rolling a die the first time does not affect the probability for rolling a die the second time, third time, and so on.
- 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 in determining all possible outcomes for multiple independent events and multiple 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 , and the probability that the sum of two dice is not 2 is . The odds in favour of rolling a sum of 2 is : or 1:35, since the fractions are both relative to the same whole.

To determine the theoretical probabilities of an event, have students first create a tree diagram or make an organized list to determine all possible outcomes and then identify the outcomes that are of interest.

For example, below are all possible outcomes for tossing one coin, three times, where the outcomes of interest are tossing two tails and one head: HTT, THT, and TTH. The theoretical probability for tossing two tails and one head is .

Have students design two probability simulations in which two to five numbers from 1 to 15 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 event).

Have students determine the experimental probabilities of only odd numbers being generated by each of the simulations and compare these to the theoretical probabilities for each simulation.