D. Data · Two-Variable Data, Scatter Plots and Probability

Multiple Dependent Events: Without Replacement

D2.2 D2.1About 55 minutes

Learning goals

I can tell when events are dependent and how the probabilities change after each event.
I can use a tree diagram with probabilities on the branches for two or more dependent events.
I can find the probability of three dependent events by multiplying the changing probabilities.
I can compare theoretical and experimental probabilities for dependent events.

Words to know

dependent events
Events where the result of one changes the probability of the next, like drawing cards without putting them back.

without replacement
An item that is picked is kept out, so there are fewer items for the next pick.

with replacement
An item that is picked is put back, so the next pick has the same choices (independent events).

Remember from Grade 7

In Grade 7, Ava picked two marbles without replacement. The total went from 5 to 4 for the second pick, so the events were dependent. You counted outcomes in a table and multiplied along tree branches. Grade 8 adds probabilities that change on each branch, and three or more dependent events.

A bag that changes

A bag has 5 red (R) and 3 blue (B) marbles: 8 in all. You pick one, keep it out, then pick another.

First pick: P(R) = 5/8 and P(B) = 3/8.
If the first was red, the bag now has 4 red and 3 blue (7 marbles). P(R) = 4/7, P(B) = 3/7.
If the first was blue, the bag now has 5 red and 2 blue (7 marbles). P(R) = 5/7, P(B) = 2/7.

The second-pick probabilities depend on the first pick. Write the probability on each branch, then multiply along each path.

Tree diagram for two picks without replacement from 5 R and 3 B. First branches: 5/8 and 3/8. After R: 4/7 and 3/7. After B: 5/7 and 2/7. Path probabilities: RR 20/56, RB 15/56, BR 15/56, BB 6/56. They add to 56/56 = 1.

Worked example

Example 1: Two picks

Use the tree. Find (a) P(both red), (b) P(one of each colour), (c) P(both red) with replacement, and compare.

(a) P(RR) = 5/8 × 4/7 = 20/56 = 5/14 (about 35.7%).
(b) One of each can happen two ways: RB or BR. 15/56 + 15/56 = 30/56 = 15/28 (about 53.6%).
(c) With replacement, the bag doesn't change: 5/8 × 5/8 = 25/64 (about 39.1%).
Without replacement, P(RR) is lower, because one red marble is already gone for the second pick.

(a) 5/14, (b) 15/28, (c) 25/64. Keeping the first marble out makes two reds less likely.

Worked example

Example 2: Three cards

Take 4 red and 2 black cards from a deck and shuffle them face down. Draw 3 cards without replacement. Find P(all 3 red) and P(at least 1 black).

First draw: 4 red of 6 cards. P = 4/6.
Second draw (if the first was red): 3 red of 5 cards. P = 3/5.
Third draw (if both were red): 2 red of 4 cards. P = 2/4.
Multiply: 4/6 × 3/5 × 2/4 = 24/120 = 1/5 (20%).
P(at least 1 black) = 1 − P(all red) = 1 − 1/5 = 4/5 (80%).

P(all 3 red) = 1/5; P(at least 1 black) = 4/5. With replacement it would be (4/6)³ = 8/27 ≈ 29.6%, so dependence really matters here.

Each red card drawn is kept out, so the number of red cards and the total both go down by 1 before the next draw: 4/6, then 3/5, then 2/4.

Worked example

Example 3: Choosing class reps

Five students, Ana, Ben, Cy, Dev and Eli, put their names in a hat. The first name drawn becomes president, the second vice-president, the third secretary. Names are not put back. (a) How many outcomes? (b) P(Ana president, Ben vice-president, Cy secretary)? (c) P(Ana, Ben and Cy get the three jobs in any order)?

(a) 5 choices, then 4, then 3: 5 × 4 × 3 = 60 outcomes.
(b) Only one of the 60 outcomes matches exactly: P = 1/60. (Or: 1/5 × 1/4 × 1/3 = 1/60.)
(c) Ana, Ben and Cy can be arranged in 3 × 2 × 1 = 6 orders (ABC, ACB, BAC, BCA, CAB, CBA), so P = 6/60 = 1/10.
Check (c) another way: first name is one of the three: 3/5. Then one of the other two: 2/4. Then the last one: 1/3. 3/5 × 2/4 × 1/3 = 6/60 = 1/10. ✓

(a) 60, (b) 1/60, (c) 1/10.

Who	Trials	All 3 red	Experimental probability	Theoretical
Leah	50	11	11/50 = 22%	1/5 = 20%
Whole class	300	63	63/300 = 21%	1/5 = 20%

Results of the 3-card experiment (4 red, 2 black, no replacement).

Worked example

Example 4: Comparing results

Leah did the 3-card experiment 50 times and got all red 11 times. Compare with the theory. What might explain a big difference if one happened?

Leah: 11/50 = 22%. Theory: 20%. Very close for 50 trials.
The class: 63/300 = 21%, closer still.
A big difference could come from not shuffling well, putting a card back by mistake (which would make it independent and raise the chance to about 30%), or recording errors.

Leah's 22% and the class's 21% both support the theoretical 20%. If results were near 30%, check whether cards were being replaced.

Steps for multiple dependent events

Decide: is each item kept out? Then the events are dependent.
For each draw, count what's left: the total goes down by 1, and the category drawn goes down by 1.
Write the probability on each branch.
Multiply along a path for "this and then this".
Add paths for "this way or that way".
Check: all path probabilities add to 1. Use 1 − P(none) for "at least one".

Just for fun

Why is a cookie jar the perfect place to learn about dependent events?

Every cookie you take changes the odds for the next person who reaches in.

Hands-on activity

Three-card experiment

You need: a deck of cards (use 4 red and 2 black cards), paper and pencil.

Work out the theory: P(all 3 red) = 4/6 × 3/5 × 2/4 = 1/5.
Shuffle the 6 cards face down. Draw 3 cards without putting any back. Record "all red" or "not all red".
Put all 6 cards back together, shuffle and repeat. Do 30 trials.
Find your experimental probability and compare with 20%.
Now try it with replacement (put each card back and shuffle before the next draw) for 30 trials. Compare with 8/27 ≈ 30%.

What to look for: about 6 of 30 trials all red without replacement, and about 9 of 30 with replacement. Results will vary, especially with only 30 trials. Pool data with a family member to get closer.

Your turn

Bag: 5 red, 3 blue. Two picks without replacement. Find P(both blue).

A jar has 4 oatmeal and 6 chocolate chip cookies. You take 2 without looking (and keep them). Find P(both oatmeal).

4 red and 2 black cards. Draw 2 without replacement. Find P(both black).

With the same 6 cards, what is P(all 3 cards drawn are black)? Explain.

Six students' names are in a hat for president, vice-president and secretary (no replacement). How many outcomes?

In 60 trials of the 3-card experiment, all red came up 13 times. Find the experimental probability and compare.

More practice: warm-up

Drawing cards and keeping them out: independent or dependent?

A bag has 8 marbles. One is taken out and kept. How many are left for the second pick?

Bag: 5 red, 3 blue. After a red is taken out, what is P(red) next?

Along a path of a tree diagram, do you add or multiply?

If P(all red) = 1/5, what is P(at least one not red)?

More practice: on your own

Two cards are drawn from a standard 52-card deck without replacement. Find P(both aces).

Two cards are drawn from a standard deck without replacement. Find P(both hearts).

A bag has 3 red and 2 blue marbles. Three are drawn without replacement. Find P(all red).

In the same bag, find P(at least one blue) in the three draws.

Jo did the 3-red experiment from the bag above 20 times and got all red 4 times (20%). The theory says 10%. Give two possible reasons.

Challenge

4 red and 2 black cards, 3 drawn without replacement. Find P(exactly 2 red).

A school raffle has 100 tickets and you bought 5. Two prize winners are drawn without replacement. Find P(you win both prizes).

A bag has 500 red and 500 blue marbles. Compare P(two reds) with and without replacement. Why is the difference so small?

Check your understanding

Quiz

Bag: 5 red, 3 blue. Two picks without replacement. P(both red) = ?

Which situation involves dependent events?

4 red and 2 black cards; draw 3 without replacement. P(all red) = ?

Five names, three jobs (no replacement). How many outcomes?

For "RB or BR", you should...

If cards were put back by mistake in the 3-card experiment, results would most likely be...

For parents and teachers

Big idea: The Ontario curriculum (D2.2) asks Grade 8 students to determine and compare the theoretical and experimental probabilities of multiple dependent events, using tools such as tree diagrams (D2.1). The key idea: when items are kept out, the totals change after every draw, so the probabilities change too.

Common difficulties:

Forgetting to reduce both the total and the category after a draw (using 5/8 × 5/8 or 5/8 × 4/8).
Multiplying when they should add (for "one of each", add the RB and BR paths).
Forgetting both orders (RB and BR).

How to help: Use a handful of real objects (coins, candies, cards) and act it out: "You took a red one. What's left now?" The card experiment with and without replacement shows the difference clearly.