Expected Outcomes and Experiments – Likelihood of Events (Primary 6 Mathematics) | Ganzaa

Likelihood of Events

Lessons

Primary 6 · Mathematics

Likelihood of Events

3 lessons

Lesson 3 of 3

Expected Outcomes and Experiments

16 min

Let's begin

Gatete says his soda bottle top is "lucky": when he flips it, it lands face up more often than face down. Habimana does not believe him. The two sides of a bottle top are different shapes, so they may not be equally likely.

How can they settle the argument? They cannot count equally likely outcomes. They have to do an experiment.

Expected outcomes

If you know the probability of an event, you can predict roughly how often it will happen in many tries:

expected number of times = probability × number of tries

Worked example 1

a fair coin is tossed 60 times. P(heads) = 12, so we expect about 12 × 60 = 30 heads.

Worked example 2

a fair die is rolled 120 times. P(6) = 16, so we expect about 120 ÷ 6 = 20 sixes.

Notice the word about. Real results wobble: in 60 tosses you might get 27 heads or 33. The more times you repeat the experiment, the closer the results usually come to what you expect.

Using an experiment to estimate a chance

When the outcomes are not equally likely, repeat the experiment many times and use the results:

estimated probability = number of times the event happened ÷ number of tries

Worked example 3

Gatete flips his bottle top 40 times. It lands face up 28 times.

Estimated probability = 2840 = 710 = 0.7.
0.7 is more than 12, so landing face up is likely for this bottle top. Gatete was right.
To be more confident, they could do 100 or 200 flips.

More likely or less likely?

To compare two chances, write both probabilities as decimals (or as fractions with the same denominator), then compare.

Worked example 4

Bag A has 3 red balls out of 5. Bag B has 5 red balls out of 10. Which gives a better chance of red?

Bag A: 35 = 0.6.
Bag B: 510 = 0.5.
0.6 is bigger, so Bag A is more likely to give red, even though Bag B has more red balls.

Chance has no memory

A fair coin has landed tails 5 times in a row. Is heads "due"? No. The coin does not remember. On the next toss, heads still has a probability of exactly 12.

See it for yourself

Open the Plinko Probability simulation and use the first screen. Drop one ball at a time: you cannot predict where each ball will land. Then drop many balls: a clear pattern builds up. One try is unpredictable; many tries settle down.

Tip

Expected number = probability × number of tries. With a few tries, anything can happen. With many tries, the results settle close to what you expect.

✦

Try it yourself

A fair coin is tossed 100 times. About how many heads do you expect?
A fair die is rolled 30 times. About how many 5s do you expect?
A spinner landed on red 18 times in 60 spins. Estimate P(red). Is red likely or unlikely?
Which gives a better chance of red: 2 red balls out of 4, or 3 red balls out of 8?

Answers

12 × 100 = 50.
30 ÷ 6 = 5.
1860 = 310 = 0.3, which is less than 12, so red is unlikely.
24 = 0.5 and 38 = 0.375, so 2 red out of 4 is better.

Quick check

A fair die is rolled 60 times. About how many times do you expect to roll a 3?

What you learned today

Expected number of times = probability × number of tries, but real results vary a little.
An experiment repeated many times lets you estimate a probability: times it happened ÷ tries.
Compare chances by comparing probabilities; past results do not change the next one.

Sources and credits

IM 6–8 Math, Grade 7, Unit 8, Lesson 4: Estimating Probabilities Through Repeated Experiments by Illustrative Mathematics (originally developed by Open Up Resources), licensed under CC BY 4.0. Adapted by Ganzaa.
Plinko Probability (version 1.1.3). Simulation by PhET Interactive Simulations, University of Colorado Boulder, licensed under CC BY 4.0 (https://phet.colorado.edu)

The bottle-top experiment and the two-bags example were written by Ganzaa.

1Expected outcomes

2Using an experiment to estimate a chance

3More likely or less likely?

4Chance has no memory

5See it for yourself