Snapdragon and
Class Height Example

Two experiments described in a common framework: the variable, its data type, the outcome space, the random variable and the measurement space. The first experiment has four possible outcomes, and the second has 150, one for each student in the class.

The experiment

In snapdragons, flower colour shows incomplete dominance: RR plants are red, WW plants are white and RW plants are pink. A lab student crosses two pink (RW) plants, grows the seeds, and records the flower colour of each offspring plant. Which allele each parent passes on is random — it happens out of sight, during meiosis and fertilisation.

The setup

Observational unitOne offspring plant

VariableFlower colour

Data typeCategorical (nominal)

Outcome space O{(R,R), (R,W), (W,R), (W,W)}
written as (allele from parent 1, allele from parent 2) — 4 equally likely outcomes

Random variable YY = the flower colour of the offspring plant
Y(R,R) = red, Y(R,W) = Y(W,R) = pink, Y(W,W) = white

Measurement space{red, pink, white} — 3 possible values

From outcome space to random variable

4 outcomes, but only 3 values. (R,W) and (W,R) both give a pink plant. In the greenhouse the student can't tell them apart — they only ever record the value of Y, never the outcome itself.

The distribution of the random variable Y

Value of Y	Outcomes that give it	Pr(Y = y)
red	(R,R)	1/4 = 0.25
pink	(R,W), (W,R)	2/4 = 0.50
white	(W,W)	1/4 = 0.25
Total	4 outcomes	1

Run the experiment

Number of offspring plants grown

Outcomes (hidden biology)

Values of Y (what the student records)

Theoretical Observed

Grow more plants and the observed proportions settle towards 1/4, 1/2, 1/4.

Questions

In snapdragons, flower colour shows incomplete dominance: plants with two R alleles have red flowers, plants with two W alleles have white flowers, and plants with one of each have pink flowers. A student crosses two pink-flowered plants and grows the resulting seeds. Each offspring receives one allele at random from each parent.

For each question, work out the answer first and say which probability rule you used and why. Then reveal the answer.

The experiment

Assume a class has 150 students, and each student’s height has been measured to the nearest cm. One student is chosen at random — every student is equally likely to be picked — and their height is recorded.

The setup

Observational unitOne student

VariableHeight

Data typeNumerical (continuous) — height is a measurement; here it is recorded to the nearest cm

Outcome space O{Student 1, Student 2, …, Student 150}
the 150 students in the class — 150 equally likely outcomes

Random variable YY = the height (cm) of the chosen student

Measurement space

The class: outcome space and random variable

Each tile is one outcome (a student). The large number is the value of Y for that outcome — the student’s height in cm.

Every student has probability 1/150 of being chosen. So the probability of any event is simply the number of students in the event ÷ 150.

Summary of the class heights

Frequency distribution of Y

Bin width = 10 cm

Each interval includes its lower boundary but not its upper one: for example, 150 – 160 means 150 ≤ height < 160.

Because every student is equally likely to be chosen, the relative frequency of an interval is exactly the probability that Y falls in it.

Questions

Assume a class has 150 students whose heights are summarised in the frequency distribution table above (bin width 10 cm). A student is chosen at random, so every student is equally likely to be picked. Use only the frequency table to answer these questions.

For each question, work out the answer first and say which probability rule you used and why. Then reveal the answer.