## D2. Probability

### Specific Expectations

#### Probability

D2.1

use fractions, decimals, and percents to express the probability of events happening, represent this probability on a probability line, and use it to make predictions and informed decisions

**probability line displaying the****likelihood****of events for an experiment involving the rolling of two six-sided dice:**

- The probability of events has numeric values ranging from 0 to 1, and percent values ranging from 0% to 100%.
- Fractions and decimals can be used to express the probability of events across the 0 to 1 continuum.

*Note*

- Have students make connections between words to describe the likelihood of events (i.e., “impossible”, “unlikely”, “equally likely”, “likely”, and “certain”) and possible fractions, decimals, and percents that can be used to represent those benchmarks on the probability line.

Have students prepare a benchmark probability line that they can use to help them make sense of numeric representations of probability and to help them make predictions about the likelihood of events. Have them mark 0 and 0% on the probability line in the same position as “impossible”, and 1 and 100% on the probability line in the same position as “certain”. Next, ask them to place on the line three fractions, decimals, and percents that could be used to represent “equally likely”, “unlikely”, and “likely”. Support students in making connections between fractions, decimals, and percents and the words used to describe the likelihood of events.

D2.2

determine and compare the theoretical and experimental probabilities of two independent events happening

**probability experiments involving two independent events:**- roll two dice 20 times
- toss two coins 15 times
- roll one die and toss one coin 30 times

**determining the theoretical probability of two independent events:**- rolling two six-sided dice:
- combinations – (first die, second die)
- (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
- (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
- (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
- (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
- (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
- (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)

- There are 36 possible outcomes.
- The theoretical probability of rolling a sum of 2 is or approximately 3%.
- The theoretical probability of rolling a sum less than 4 is or approximately 8% because there are three possibilities: (1, 1) (1, 2) (2, 1).

- combinations – (first die, second die)

- rolling two six-sided dice:
**comparing experimental probability with theoretical probability:**- The more trials done in an experiment, the closer the experimental probability will be to the theoretical probability. Increase the number of trials by:
- continuing the experiment over multiple trials, or
- combining classmates’ experimental results.

- The more trials done in an experiment, the closer the experimental probability will be to the theoretical probability. Increase the number of trials by:

- 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.
- The more trials completed in an experiment, the closer the experimental probability will be to the theoretical probability.
- The sum of the probabilities 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.

*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.

Before students conduct a probability experiment, it is important for them to identify all the possible outcomes, which are the combinations of the two independent events. Have them draw a tree diagram to show all the possible outcomes for tossing one coin and rolling one die. They can use abbreviations to simplify the list of outcomes. For example, tossing a tail and rolling a 1 could be recorded as T1.

Ask students to determine the probability of various kinds of events based on experimental data and compare it to the theoretical probability. Ask questions such as:

- What is the probability of tossing a head and rolling an even number?
- What is the probability of tossing a head and not rolling a 3?