Random variables and probability tables | Year 12

01 / LEARN

The idea, step by step

A number attached to an outcome

A random variable assigns one numerical value to each outcome of an uncertain process. Use a capital letter for the variable and a lower-case letter for a possible value. The variable is discrete when its possible values can be listed, even if the list is infinite. A count of late buses is discrete; an exact waiting time is normally modelled as continuous.

Define the measurement and the observation period before making a table. If X counts heads in two coin tosses, the values are 0, 1 and 2. The outcomes HT and TH are different outcomes but give the same value X = 1.

A complete table carries all the probability

The values in the first column must be distinct and exhaustive. The corresponding probabilities cannot be negative and must add to 1. A missing value can have probability zero, but an outcome that is actually possible cannot simply be ignored.

A probability bar chart places a separate bar over each discrete value. The bar heights, not the spaces between bars, give the probabilities. This differs from the area interpretation used later for a continuous density.

0 \leq P (X = x) \leq 1, \sum_{x} P (X = x) = 1

A common trap

Every possible value is equally likely because it is listed once.

Instead: A table groups underlying outcomes. Count or model the probability of each group; listing a value once does not make its probability equal to the others.

02 / WORKED EXAMPLES

See the thinking

Example 1

Two tosses, three values

Two independent fair coins are tossed. Let X be the number of heads. Construct its distribution.

The four equally likely outcomes are TT, TH, HT and HH.
TT gives 0 heads; TH and HT give 1 head; HH gives 2 heads.
Count the outcomes in each group and divide by 4. Check that the probabilities sum to 1.

P (X = 0) = 1 / 4, P (X = 1) = 1 / 2, P (X = 2) = 1 / 4.

Example 2

Find a normalising constant

X has values 1, 2 and 3 with probabilities $k, 2 k, 3 k$ . Find k and P(X = 3).

$k + 2 k + 3 k = 1$ , so $6 k = 1$ .
$k = 1 / 6$ . Each resulting probability is between 0 and 1.
$P (X = 3) = 3 k = 1 / 2$ .

k = 1 / 6; P (X = 3) = 1 / 2.

Check your understanding

A probability table has entries 0.15, 0.45 and p. Find p.

Show answer

p = 0.40.

The mutually exclusive exhaustive probabilities must sum to 1, so p = 1 − 0.15 − 0.45.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Let X count the number of goals in a match. Explain why X is discrete.
1 mark
X takes values 0, 2 and 5 with probabilities 0.2, 0.5 and p. Find p.
2 marks
For values −1, 0 and 1, probabilities are $k, 4 k, 3 k$ . Find k and the probability of a positive value.
3 marks
A proposed table uses probabilities 0.4, −0.1 and 0.7. Is it valid? Explain.
2 marks
A bag contains three red and one blue counter. One is selected fairly. Define X = 1 for red and X = 0 for blue. Give its distribution.
2 marks
Two fair independent coin tosses are made. Let Y be the number of tails. Give all values and probabilities.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Let X count the number of goals in a match. Explain why X is discrete.
1 mark
Goals are counted in whole numbers.
The possible values can be listed.
Its possible values are 0, 1, 2, … .
X takes values 0, 2 and 5 with probabilities 0.2, 0.5 and p. Find p.
2 marks
Add the known probabilities: 0.2 + 0.5 = 0.7.
Use total probability 1: p = 1 − 0.7 = 0.3.
p = 0.3.
For values −1, 0 and 1, probabilities are $k, 4 k, 3 k$ . Find k and the probability of a positive value.
3 marks
$8 k = 1$ , so $k = 1 / 8$ .
Only the value 1 is positive, giving probability 3k.
$k = 1 / 8; P (X > 0) = 3 / 8$ .
A proposed table uses probabilities 0.4, −0.1 and 0.7. Is it valid? Explain.
2 marks
The entries do sum to 1.
That is insufficient: every individual probability must be at least zero.
No. A probability is negative.
A bag contains three red and one blue counter. One is selected fairly. Define X = 1 for red and X = 0 for blue. Give its distribution.
2 marks
There are four equally likely counters.
Group the three red outcomes at X = 1 and the one blue outcome at X = 0.
$P (X = 0) = 1 / 4, P (X = 1) = 3 / 4$ .
Two fair independent coin tosses are made. Let Y be the number of tails. Give all values and probabilities.
3 marks
List HH, HT, TH and TT.
The tail counts are 0, 1, 1 and 2; group equal counts.
$y = 0, 1, 2$ with probabilities $1 / 4, 1 / 2, 1 / 4$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Mathematical Methods: discrete random variables and probability distributions, expected value and variance, transformations and independent sums, Bernoulli and binomial models. Original supporting lessons and exercises follow the approved Ascent chapter structure.

Official curriculum source ↗

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