Topic 7 of 7

Probability and sampling

Step 9 · Proportional reasoning topics

Give probabilities as numbers from 0 to 1, find them by counting equally likely outcomes or by experiment, predict expected counts, and use random samples to estimate.

Probability from 0 to 1

Probability measures how likely something is, from 0 (impossible) to 1 (certain). When all outcomes are equally likely, the probability is the number of ways it can happen divided by the number of outcomes.

0 is impossible, 0.5 is an even chance, and 1 is certain.

Example: Rolling a number cube

A number cube shows 1 to 6. What is the probability of rolling an even number?

Even numbers: 2, 4 and 6, so 3 ways.
6 outcomes in all.
$\frac{3}{6}$ = $\frac{1}{2}$ .

Answer $\frac{1}{2}$

Example: Not happening

A bag has 2 red, 6 blue and 2 green marbles. What is the probability of not picking blue?

10 marbles, and 4 of them are not blue.
$\frac{4}{10}$ = $\frac{2}{5}$ .
Or: 1 minus the chance of blue, 1 − $\frac{6}{10}$ .

Answer $\frac{2}{5}$

Experiments and expected counts

When outcomes cannot be counted, run an experiment. The experimental probability is how often it happened divided by the number of tries. The more tries, the closer it tends to get to the true probability.

50 rolls of a number cube: each number came up close to 50 ÷ 6, about 8 times.

Example: Experimental probability

A spinner landed on red 18 times in 60 spins. What is the experimental probability of red?

$\frac{18}{60}$ = $\frac{3}{10}$ .

Answer $\frac{3}{10}$ , or 0.3

Example: Expected count

The probability of rain on a day is 0.2. About how many rainy days should you expect in 30 days?

0.2 × 30 = 6.

AnswerAbout 6 days

Samples

To learn about a big group (the population), we often ask a smaller group (a sample). In a random sample everyone has the same chance of being picked, so the sample tends to look like the population.

Example: Estimating from a sample

In a random sample of 50 students at a school of 800, 15 walk to school. About how many walk in the whole school?

$\frac{15}{50}$ = 0.3 of the sample walk.
0.3 × 800 = 240.

AnswerAbout 240 students

Different random samples give slightly different answers. Taking several samples shows how much an estimate can vary.

Try it yourself

Work each one on paper first, then open the answer to check.

A number cube is rolled. What is the probability of a number greater than 4?
Show answer
$\frac{2}{6}$ = $\frac{1}{3}$
A bag has 4 red and 6 yellow counters. What is the probability of red?
Show answer
$\frac{4}{10}$ = $\frac{2}{5}$
A coin is flipped 200 times. About how many heads should you expect?
Show answer
About 100
A player makes 70% of free throws. In 40 shots, about how many go in?
Show answer
About 28
In a random sample of 40 voters, 24 want a new park. Out of 2,000 voters, about how many want it?
Show answer
About 1,200
Is asking people at a gym how often they exercise a fair sample of a town?
Show answer
No. People at a gym exercise more than most, so the sample is biased.