Probability models and assumptions

The idea

A probability model rests on assumptions, such as equally likely outcomes or independent trials. The probabilities of all outcomes must add to $1$ . Judge a model by whether its assumptions fit the real situation.

Check the probabilities add to $1$ and use that to find unknowns.
Expected frequency $=$ probability $\times$ number of trials.
Question each assumption in context: could one outcome affect the next?

Watch out Independence must come from the situation, not from convenience.

See it worked

Worked example

A model for a spinner gives $P(\text{red}) = 0.25$ . It is spun $80$ times and lands on red $31$ times. Comment on the model.

Expected reds: $0.25 \times 80 = 20$ .
$31$ is well above $20$ .
The model may be wrong; more spins would test it.

Answer Expected $20$ reds but saw $31$ : the model may be unsuitable

Your turn

Question 1Non-calculator3 marks

A discrete random variable takes four values with probabilities $0.2$ , $0.35$ , $k$ and $2k$ . Find $k$ .

Question 2Non-calculator3 marks

A coin is modelled with $P(\text{heads}) = 0.5$ . It is tossed $200$ times. How many heads does the model predict?

Question 3Non-calculator3 marks

A model treats whether it rains on each day of a week as independent, each with probability $0.3$ . Give one reason this assumption may be unrealistic.

Write your answer, then show the working

Independence means one day's weather tells you nothing about the next.
In reality, rainy spells last several days.
So rain on one day makes rain the next day more likely.

Answer Weather on one day affects the next, so the days are not independent

In past papers

10 official questions in 8 papers since 2018. The latest:

All A-level Maths past papers

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