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

Multiple Independent Events: Tree Diagrams, Theory and Experiments

D2.2 D2.1 A1About 55 minutes

Learning goals

I can use a tree diagram to list the outcomes of three or more independent events.
I can find the total number of outcomes by multiplying.
I can find the theoretical probability of multiple independent events by multiplying probabilities.
I can compare theoretical and experimental probabilities and explain differences.

Words to know

independent events
Events where the result of one does not change the probability of the others, like flipping several coins.

tree diagram
A diagram with branches that shows every possible outcome of a sequence of events.

theoretical probability
The probability worked out by reasoning: favourable outcomes ÷ total equally likely outcomes.

experimental probability
The probability found from an experiment: number of times the event happened ÷ number of trials.

at least one
One or more. Often easiest to find as 1 − P(none).

Remember from Grade 7

In Grade 7 you found the theoretical probability of two independent events (with a table or tree diagram) and compared it with experimental probability. You also saw that you can multiply along the branches of a tree. Grade 8 extends this to three or more events, where tables no longer work but trees and multiplication do.

Three coins at once

Before a pick-up game of road hockey, three friends each flip a coin. What is the probability that all three land heads?

A table only has two directions (rows and columns), so it can't show three coins easily. A tree diagram can: each coin adds another set of branches.

Tree diagram for flipping 3 coins. The first coin splits into H and T, each of those splits again for the second coin, and again for the third. The 8 outcomes (right) are HHH, HHT, HTH, HTT, THH, THT, TTH and TTT, with the number of heads beside each.

Counting outcomes by multiplying

Each coin has 2 outcomes, so 3 coins have 2 × 2 × 2 = 8 outcomes. In general, for independent events, multiply the number of outcomes of each event. A coin, a die and a spinner with 3 equal sections have 2 × 6 × 3 = 36 outcomes.

Multiplying probabilities

For independent events, P(A and B and C) = P(A) × P(B) × P(C).

P(HHH) = 1/2 × 1/2 × 1/2 = 1/8. That matches the tree: 1 outcome out of 8.

Number of heads	Outcomes	Count	Theoretical probability
0	TTT	1	1/8 = 12.5%
1	HTT, THT, TTH	3	3/8 = 37.5%
2	HHT, HTH, THH	3	3/8 = 37.5%
3	HHH	1	1/8 = 12.5%
Total		8	8/8 = 100%

Sorting the 8 outcomes of 3 coins by the number of heads.

Worked example

Example 1: Exactly two and at least one

Three coins are flipped. Find (a) P(exactly 2 heads) and (b) P(at least 1 head).

(a) From the tree: HHT, HTH and THH have exactly 2 heads. P = 3/8.
(b) "At least 1 head" means 1, 2 or 3 heads: 3 + 3 + 1 = 7 outcomes, so 7/8.
Faster way for (b): the only outcome with no heads is TTT. P(at least 1 head) = 1 − P(no heads) = 1 − 1/8 = 7/8.

(a) 3/8 = 37.5%, (b) 7/8 = 87.5%.

Worked example

Example 2: Coin, die and spinner

Flip a coin, roll a standard die and spin a spinner with 3 equal sections labelled A, B and C. Find (a) the number of outcomes, (b) P(heads, 6, A), (c) P(heads, even number, not C).

(a) 2 × 6 × 3 = 36 outcomes.
(b) P(H) × P(6) × P(A) = 1/2 × 1/6 × 1/3 = 1/36.
(c) P(H) = 1/2, P(even) = 3/6 = 1/2, P(not C) = 2/3. Multiply: 1/2 × 1/2 × 2/3 = 2/12 = 1/6.
Check (c) by counting: 1 × 3 × 2 = 6 favourable outcomes out of 36 = 1/6. ✓

(a) 36, (b) 1/36, (c) 1/6.

Worked example

Example 3: Guessing on a quiz

A quiz has 3 multiple-choice questions, each with 4 choices. Raj guesses every answer. Find P(all 3 correct) and P(at least 1 correct).

Each guess is independent. P(correct) = 1/4 and P(wrong) = 3/4.
P(all correct) = 1/4 × 1/4 × 1/4 = 1/64 (about 1.6%).
P(all wrong) = 3/4 × 3/4 × 3/4 = 27/64.
P(at least 1 correct) = 1 − 27/64 = 37/64 (about 58%).

P(all correct) = 1/64; P(at least 1 correct) = 37/64. Guessing is a poor strategy for getting everything right!

Theory meets experiment

Kai flipped 3 coins 80 times and recorded the number of heads. Then the whole class pooled their results: 800 trials.

Number of heads	Kai: 80 trials	Kai: experimental	Class: 800 trials	Class: experimental	Theoretical
0	9	11.25%	97	12.1%	12.5%
1	31	38.75%	305	38.1%	37.5%
2	29	36.25%	298	37.3%	37.5%
3	11	13.75%	100	12.5%	12.5%
Total	80	100%	800	100%	100%

Experimental probability = count ÷ number of trials. Class percentages are rounded to one decimal place. The class results, with 10 times as many trials, are closer to the theory.

The same numbers as the table. Kai's 80 trials wobble around the theory; the class's 800 trials are almost exactly the height of the theoretical bars.

Worked example

Example 4: Comparing

Compare Kai's experimental P(3 heads) with the theoretical probability. Then predict how many times you'd expect 3 heads in 200 trials.

Kai: 11/80 = 13.75%. Theory: 1/8 = 12.5%. Kai's result is a little high, which is normal with only 80 trials.
The class's 800 trials gave 100/800 = 12.5%, right on the theory.
Prediction: 1/8 × 200 = 25 times. Real results will usually be close to 25, but not exact.

Kai's 13.75% is close to 12.5%; more trials (the class's 800) came even closer. Expect about 25 sets of 3 heads in 200 trials.

Why experiments and theory differ

Chance: a small number of trials can be far from the theory just by luck. More trials usually get closer.
Unfair equipment or method: a bent coin, a spinner that sticks, or flips that aren't high enough.
Recording mistakes.

If many trials are still far from the theory, check the equipment, the method and the theory. Coins have no memory: after TTT, the next flip is still 1/2 heads.

Just for fun

If everyone in a class of 30 flips 3 coins once, probability says about 4 people will get HHH, and every one of them will feel like a genius. Somebody had to! (1/8 × 30 = 3.75.)

Hands-on activity

Three-coin experiment

You need: 3 coins (any kind), a cup to shake them in, paper and pencil.

Copy the table with rows for 0, 1, 2 and 3 heads.
Shake the 3 coins in the cup, tip them out and record the number of heads with a tally mark. Do 40 trials.
Find your experimental probability for each row (count ÷ 40). Compare with 1/8, 3/8, 3/8 and 1/8.
If a family member does 40 more trials, combine the results (80 trials). Are they closer to the theory?
Bonus: add a die. How many outcomes are there now? What is P(3 heads and a 6)?

What to look for: 1 and 2 heads happen most often (about 15 times each in 40 trials) and 0 and 3 heads least often (about 5 times each). Bonus: 2 × 2 × 2 × 6 = 48 outcomes; P(3 heads and a 6) = 1/8 × 1/6 = 1/48.

Your turn

How many outcomes are there when you flip 4 coins?

Find P(all heads) when 4 coins are flipped.

A spinner with 3 equal sections is spun and 2 coins are flipped. How many outcomes?

Three standard dice are rolled. Find P(all three show 6).

Three coins are flipped. Find P(exactly 1 head).

Kai got 0 heads in 9 of 80 trials. What is the experimental probability? Compare with the theory.

About how many times would you expect exactly 2 heads in 400 flips of 3 coins?

More practice: warm-up

Are the results of 3 separate coin flips independent?

How many outcomes for a coin and a die?

P(heads) × P(heads) × P(heads) = ?

If P(none) = 1/8, what is P(at least one)?

Which is usually closer to the theoretical probability: 20 trials or 2 000 trials?

More practice: on your own

Four coins are flipped. Find P(at least one tail).

A spinner has 4 equal sections A, B, C, D. It is spun 3 times. Find P(A all three times) and P(no A at all).

Flip a coin, roll a die and spin the A-B-C spinner. Find P(tails, a number less than 3, B).

Kai's friend did 16 trials of 3 coins and got 3 heads 5 times. She says, "P(3 heads) must be about 31%." Respond.

A camp offers 3 breakfasts, 4 lunches and 2 dinners, each chosen at random for the day. How many different daily menus are possible? What is P(getting your favourite of each)?

Challenge

Raj guesses on 3 questions with 4 choices each. Find P(exactly 2 correct).

What is the smallest number of coins you must flip so that P(all heads) is less than 1%?

Use the table for 3 coins to check that the probabilities of 0, 1, 2 and 3 heads add to 1. Then explain why P(at least 1 head) = 1 − P(0 heads) always works.

Check your understanding

Quiz

How many outcomes when you flip 3 coins?

P(HHH) with 3 coins is...

A coin, a die and a 3-section spinner give how many outcomes?

Three coins: P(at least one head) = ?

In 80 trials, 3 heads came up 11 times. The experimental probability is...

Why are the class results (800 trials) closer to the theory than Kai's (80 trials)?

Three questions, 4 choices each, all guessed. P(all correct) = ?

For parents and teachers

Big idea: The Ontario curriculum asks Grade 8 students to determine and compare the theoretical and experimental probabilities of multiple independent events (D2.2), using tools such as tree diagrams (D2.1). Grade 7 handled two events; Grade 8 handles three or more.

Common difficulties:

Adding instead of multiplying the number of outcomes (2 + 6 + 3 instead of 2 × 6 × 3).
Missing outcomes like HTH when listing; a tree diagram prevents this.
Expecting experiments to match exactly, or trusting very few trials.
The gambler's fallacy: "After three tails, heads is due." Coins have no memory.

How to help: Do the 3-coin experiment together and pool your results. Ask, "How many trials do you think we'd need to get close to 12.5%?" The lesson also builds A1 skills: persevering, recording carefully and reasoning about results.