01 · Count late deliveries
Four deliveries are made. X counts how many are late. Assuming every count can occur, list the support.
Hint
Include zero and the largest possible count.
Worked solution
{0,1,2,3,4}.
Understand · explore · practise
Discrete random variables
Define a numerical random variable, list its possible values and build a probability distribution by combining the outcomes that produce the same value.
Before you startSample spaces, equally likely outcomes and independent selections.
01 / Choose what to record
A random variable assigns a number to each outcome.
The experiment produces an outcome; X records a numerical feature of it.
Capital X names the variable. A lower-case x is one possible value.
For two deliveries, an outcome could be “early, late”. If X counts late deliveries, that outcome gives X=1. Another outcome, “late, early”, also gives X=1. A variable can deliberately forget details of the original outcome.
From outcomes to a scoreExplore
Choose A uniformly from {0,1,3} and B independently and uniformly from {0,2}. Define X=A+B. Each ordered pair has probability 1/6.
X takes values 0,1,2,3,5. Their probabilities are 1/6,1/6,1/6,2/6,1/6 respectively.
02 / List the possible values
The support is the set of values with positive probability.
For the score X=A+B in the model:Worked example
A=0 gives scores 0 or 2; A=1 gives 1 or 3
B can be 0 or 2.
A=3 gives scores3 or 5
The score 3 appears in two different pairs.
Support: {0,1,2,3,5}
Do not include 4 just because it lies between the minimum and maximum.
02 · A score, not a count
A token is chosen uniformly from tokens labelled −2,0,7. X records the label. List its support.
Hint
A discrete variable need not take consecutive positive integers.
Worked solution
{−2,0,7}. Negative values and gaps are allowed.
03 / Combine outcomes with the same value
Equal pair probabilities can produce unequal score probabilities.
Watch: group six outcomes into five score values
Pause, replay or seek freely. The notes explain the same idea and stay in view.
P(X=x) is the total probability of all outcomes giving value x.
Here, each pair has mass 1/6; the score 3 receives two such masses.
The pairs (1,2) and (3,0) both give 3. So P(X=3)=2/6=1/3. Each other supported score has only one pair and probability 1/6.
03 · An impossible score
Find P(X=4) for the model.
Hint
Check which pairs produce4.
Worked solution
0. There is no eligible pair.
04 · A common error
Why is assigning probability 1/5 to each supported score wrong?
Hint
Equal values in a list are not automatically equally likely.
Worked solution
The five values are supported by unequal numbers of equally likely pairs. Score3 has two pairs; each other score has one.
04 / Read a probability distribution
A table lists values and their probability masses.
| x | P(X=x) |
|---|---|
| 0 | 1/6 |
| 1 | 1/6 |
| 2 | 1/6 |
| 3 | 2/6 |
| 5 | 1/6 |
A probability bar diagram shows the same information: the bar at 3 has height 1/3, while the other four heights are 1/6. Bars represent probability at isolated values, not density over intervals.
05 · Total mass
Check that the table accounts for the entire experiment.
Hint
Add every mass once.
Worked solution
(1+1+1+2+1)/6=1.
06 · Value versus probability
Does P(X=3)=1/3 mean that X takes the value1/3?
Hint
Separate the score from the chance of that score.
Worked solution
No. The score is 3. Its probability is 1/3. The model never gives the score 1/3.
05 / Add the values that qualify
An inequality names a set of supported values.
07 · Strict inequality
Find P(X<3) in the model.
Hint
Include 0,1,2 only.
Worked solution
1/6+1/6+1/6=1/2.
08 · Inclusive inequality
Find P(X≤3).
Hint
Include the mass at 3 as well.
Worked solution
(1+1+1+2)/6=5/6.
09 · Odd score
Find P(X is odd).
Hint
Which supported values are odd?
Worked solution
Values 1,3,5 qualify, so probability=(1+2+1)/6=2/3.
10 · A non-integer cutoff
Find P(2<X<5).
Hint
Only supported values strictly inside the interval count.
Worked solution
Only 3 qualifies. Probability 1/3. There is no mass at 4.
06 / Distinguish a count from a measurement
Discrete describes possible values, not whether a decimal is written.
A count such as number of messages has separate values. An idealised elapsed time is usually modelled as continuous. Recording time to the nearest second produces discrete recorded values; that does not make the underlying physical time a count.
Discrete values can be decimals: a random prize of £0,£1.50 or£4 is discrete. A fixed quantity can be represented by a degenerate distribution, with probability 1 at its fixed value, although it contains no uncertainty.
11 · Measured or recorded?
A duration is recorded to the nearest 0.1 second. Distinguish the physical duration and the stored reading.
Hint
Rounding restricts the stored values.
Worked solution
The idealised duration can vary continuously; the recorded reading lies on a grid of tenths and is discrete.
12 · Always the same
A sealed box always contains exactly 6 tokens. X counts them. What is its distribution?
Hint
All probability is concentrated at one value.
Worked solution
P(X=6)=1, and P(X=x)=0 for every other x. It is a constant, or degenerate, random variable.
07 / Keep the distribution tied to the experiment
A different selection rule can change the masses.
13 · Unequal first selection
Now P(A=0)=1/2, P(A=1)=1/4, P(A=3)=1/4, while B remains independent and uniform on {0,2}. Find P(X=3).
Hint
Add probabilities for(1,2) and(3,0).
Worked solution
(1/4)(1/2)+(1/4)(1/2)=1/4. The old 1/3 relied on uniform A.
14 · Predicting frequencies
In 600 independent repeats of the original model, must score 3 appear exactly 200 times?
Hint
A model probability is not a quota.
Worked solution
No. 200 is the long-run proportion applied to 600 as an expected count, not a guaranteed observed count. Random samples fluctuate.
08 / Define, list, group and check
Start from the experiment before writing a table.
Define the numerical quantity X, identify its support and combine the probabilities of outcomes with equal values. Sum supported masses for events. Check the total is 1 and keep the assumptions behind any equal-likelihood calculation explicit.
Section 1 of 8 · Choose what to record