D. Data · Data Literacy and Probability

Theoretical Probability of Two Independent or Two Dependent Events

D2.2 D2.1About 55 minutes

Learning goals

I can list all the outcomes of two events using a table or a tree diagram.
I can find the theoretical probability of two independent events.
I can find the theoretical probability of two dependent events, where the first item is not put back.
I can compare the probabilities with and without replacement and explain the difference.

Words to know

outcome
One possible result, like G1B2 (green marble 1, then blue marble 2).

sample space
The list of all possible outcomes.

theoretical probability
Number of favourable outcomes ÷ number of possible outcomes, when all outcomes are equally likely.

tree diagram
A diagram with branches that shows every outcome of two or more events.

Remember from Grade 6

To find the probability of two independent events, you listed every outcome in a tree diagram or table, then counted: P = favourable outcomes ÷ total outcomes. A coin and a 3-section spinner gave 2 × 3 = 6 outcomes.

Two picks from the marble bag

Ava's bag still has 3 green and 2 blue marbles. To count fairly, give each marble a name: G1, G2, G3, B1, B2. Each marble is equally likely to be picked.

Ava picks two marbles, one after the other. What is the probability that both are blue? The answer depends on whether she puts the first marble back.

Worked example

Example 1: With replacement (independent)

Ava picks a marble, puts it back, and picks again. Find P(both blue).

First pick: 5 choices. Second pick: still 5 choices. Total outcomes: 5 × 5 = 25. They are listed in the first table below.
Both blue: B1B1, B1B2, B2B1, B2B2. That is 4 outcomes (in bold in the table).
P(both blue) = 4/25 = 0.16 = 16%.

4/25 (16%)

1st \ 2nd	G1	G2	G3	B1	B2
G1	G1G1	G1G2	G1G3	G1B1	G1B2
G2	G2G1	G2G2	G2G3	G2B1	G2B2
G3	G3G1	G3G2	G3G3	G3B1	G3B2
B1	B1G1	B1G2	B1G3	B1B1	B1B2
B2	B2G1	B2G2	B2G3	B2B1	B2B2

With replacement: 25 outcomes. The 4 outcomes where both marbles are blue are in bold.

Worked example

Example 2: Without replacement (dependent)

Now Ava picks a marble, keeps it out, and picks again. Find P(both blue).

First pick: 5 choices. Second pick: only 4 choices, because the first marble is gone. Total outcomes: 5 × 4 = 20. In the table below, ✕ marks the impossible outcomes, like picking G1 twice.
Both blue: B1B2 and B2B1. That is only 2 outcomes.
P(both blue) = 2/20 = 1/10 = 0.1 = 10%.

1/10 (10%)

1st \ 2nd	G1	G2	G3	B1	B2
G1	✕	G1G2	G1G3	G1B1	G1B2
G2	G2G1	✕	G2G3	G2B1	G2B2
G3	G3G1	G3G2	✕	G3B1	G3B2
B1	B1G1	B1G2	B1G3	✕	B1B2
B2	B2G1	B2G2	B2G3	B2B1	✕

Without replacement: 25 − 5 = 20 possible outcomes. The 2 outcomes where both marbles are blue are in bold.

Why is it smaller without replacement?

After a blue marble comes out and stays out, only 1 blue marble is left among 4. So a second blue is less likely. That is why P(both blue) dropped from 16% to 10%.

A tree diagram with probabilities

You can also put the probability on each branch of a tree diagram. For the second pick, the branches change depending on the first pick, because the events are dependent.

Tree diagram without replacement. First pick: G with 3/5, B with 2/5. After G: G 2/4, B 2/4. After B: G 3/4, B 1/4. Outcomes: GG 6/20, GB 6/20, BG 6/20, BB 2/20.

To find the probability of one path, multiply along the branches. For two blues: 2/5 × 1/4 = 2/20, the same answer as counting in the table. The four paths add to 6/20 + 6/20 + 6/20 + 2/20 = 20/20 = 1, which is a good check.

You learned to multiply fractions in the Operations unit (B2.8). If you prefer, you can always count outcomes in a table instead. Both methods give the same answer.

With replacement, the bag is the same for the second pick, so every second branch is 3/5 or 2/5. Multiplying along the B-B path gives 2/5 × 2/5 = 4/25, matching the table.

Worked example

Example 3: One of each colour

Without replacement, what is P(one green and one blue, in either order)?

From the tree: GB is 6/20 and BG is 6/20.
Either order counts, so add: 6/20 + 6/20 = 12/20 = 3/5.
Check by counting in the table: G then B is 3 × 2 = 6 outcomes, and B then G is 2 × 3 = 6 outcomes. 12 of 20. ✓

3/5 (60%)

Worked example

Example 4: Choosing two class jobs

Four students, Ana, Ben, Cy and Dev, put their names in a hat. The first name drawn is president; that name stays out, and the second name drawn is vice-president. (a) What is P(Ana is president and Ben is vice-president)? (b) What is P(Ana gets one of the two jobs)?

Total outcomes: 4 choices for president × 3 for vice-president = 12.
(a) Only one outcome is Ana then Ben. P = 1/12.
(b) Ana as president: 3 outcomes (Ana-Ben, Ana-Cy, Ana-Dev). Ana as vice-president: 3 outcomes (Ben-Ana, Cy-Ana, Dev-Ana). 6 of 12.
P(Ana gets a job) = 6/12 = 1/2.

(a) 1/12, (b) 1/2. These are dependent events: a name drawn first can't be drawn second.

Steps for two events

Decide: independent (put back, or separate objects) or dependent (kept out)?
Find the total outcomes: first choices × second choices (for dependent events, the second number is smaller).
List or count the favourable outcomes in a table or tree diagram.
P = favourable ÷ total. Simplify, and write it as a percent if helpful.
Check that all the probabilities add to 1.

Just for fun

Tree diagrams are the only trees that grow sideways and still bear fruit. Their fruit just happens to be probabilities!

Hands-on activity

Sock drawer model

You need: 5 socks of the same size and fabric (3 dark and 2 light) and a pillowcase or bag, so you can't tell them apart without looking. Or use 5 playing cards: 3 hearts and 2 spades.

Label the items on paper: D1, D2, D3, L1, L2 (or H1, H2, H3, S1, S2 for cards).
Make the without replacement table (5 by 5, with ✕ on the diagonal). Count the outcomes where both items are light (or both are spades).
Work out P(both light) without replacement (2/20 = 1/10) and with replacement (4/25).
Ask a family member to guess which is more likely before you show them the math.

Math talk: "Why is the total 20 instead of 25 when you don't put the first item back?"

Your turn

A bag has 2 red (R1, R2) and 2 yellow (Y1, Y2) tiles. You pick two with replacement. How many outcomes are possible?

Same bag, with replacement. What is P(both red)?

Same bag, without replacement. How many outcomes are possible?

Same bag, without replacement. What is P(both red)?

Same bag, without replacement. What is P(one red and one yellow, in either order)?

You flip a coin and roll a die (independent). What is P(tails and a number greater than 4)?

Why is P(both red) smaller without replacement than with replacement?

More practice: warm-up

Two coins are flipped. How many outcomes?

Two coins are flipped. What is P(both heads)?

A spinner with 3 equal sections is spun twice. How many outcomes?

Three names in a hat. Two are drawn without putting the first back. How many ordered outcomes?

A die is rolled twice. How many outcomes?

More practice: on your own

A die is rolled twice. What is P(a 6 both times)?

Cards numbered 1, 2, 3, 4, 5. Pick two without replacement. What is P(both even)?

Same cards, picked with replacement. What is P(both even)?

A spinner has 4 equal sections, one of each: red, blue, green, yellow. Spin it twice. What is P(the same colour both times)?

From 5 students, a teacher chooses one to take attendance and then a different one to water the plants. How many ways are there? What is P(Sam takes attendance and Kim waters the plants)?

Challenge

A bag has 4 green and 1 blue marble. Two are picked without replacement. Find P(both green), P(both blue), and P(one of each). Check that they add to 1.

Use a tree diagram with probabilities: a bag has 3 green and 2 blue marbles; two are picked without replacement. Find P(green, then green) by multiplying along the branches.

A drawer has 6 socks: 4 black and 2 white. In the dark, you take out 2 socks. Is it more likely that they match or don't match?

Check your understanding

Quiz

Two marbles are picked with replacement from a bag of 5. How many outcomes are there?

Two marbles are picked without replacement from a bag of 5. How many outcomes are there?

A bag has 3 green and 2 blue marbles. Two are picked without replacement. What is P(both blue)?

A coin is flipped and a die is rolled. What is P(heads and 3)?

Cards 1, 2, 3, 4 are picked two at a time without replacement. What is P(1, then 2)?

On a tree diagram, how do you find the probability of one path?

Why does P(both blue) go down when the first marble is not put back?

For parents and teachers

Big idea: D2.2 asks Grade 7 students to determine the theoretical probabilities of two independent events and two dependent events. Naming each item (G1, G2, ...) makes every outcome equally likely and easy to count. A table or tree diagram shows the whole sample space. Multiplying along tree branches is a shortcut; counting in a table always works.

Common difficulties:

Using 5 × 5 instead of 5 × 4 when the first item is kept out.
Not giving each item its own name, so outcomes like BB look like one outcome instead of two (B1B2 and B2B1).
Forgetting both orders for "one of each".

How to help: Act it out with socks, cards or wrapped candies. Ask your child to explain why the total changes. Probabilities that add to 1 are a great check.