Home‎ > ‎Statistics prompts‎ > ‎

### Coins inquiry

P(2 heads and 2 tails) when you flip 4 coins
P(1 head and 1 tail) when you flip 2 coins

The prompt is suitable for introducing the concept of probability. It uses mathematical notation in a familiar context of flipping coins. The superficial similarity of the cases (the number of heads and tails both being half the number of coins) might lead students to speculate that the prompt is true. In aiming to set the prompt just above the level of the class, the teacher might expect the students to be able to explain the case of two coins, but four coins would be beyond their powers of reasoning. In the classroom, the prompt has developed into the following lines of inquiry:
• Listing combinations
• Carrying out practical experiments with coins or using software such as Excel to simulate the experiment (experimental probability)
• Drawing sample space diagrams
• Calculating theoretical probability
• Using probability tree diagrams
• Understanding the relevance of Pascal's triangle (combinations, patterns and the binomial expansion)
• Changing the prompt (more coins, different outcomes)
• Looking for a pattern in the probabilities of the outcomes with 2, 4, 6 ... coins.
A curious discovery
In an Inquiry Maths workshop at the University of Birmingham (December 2019), two trainee teachers, John Wood and Ben Crossley, made a curious discovery:

P(2 heads) when you flip 3 coins = P(2 heads) when you flip 4 coins

Their observation can be generalised, as John and Ben went on to show using Pascal's triangle (see their notes above).

P(1 head) when you flip 1 coin = P(1 head) when you flip 2 coins
P(3 heads) when you flip 5 coins = P(3 heads) when you flip 6 coins
P(n heads) when you flip (2n - 1) coins =
P(n heads) when you flip 2n coins

John and Ben are studying for a Postgraduate Diploma in Secondary Education. The course leader is Tom Francome who you can follow on twitter @TFrancome.

Alternative prompt
P(2 heads and 2 tails) when you flip 4 coins =
2 x P(1 head and 1 tail) when you flip 2 coins
The main prompt was inspired by this alternative, which was devised by Nicola Stokes (a Head of Mathematics in East Sussex, UK). Nicola tried it out with her year 9 class. The students responded to the prompt by asking the questions and making the statements above. Nicola reports that the prompt "worked really nicely and all of the class chose to continue to investigate if the prompt was true as they couldn't immediately see that it wasn't." During the inquiry, the students developed fluency in constructing sample space diagrams and tree diagrams.

Resources
Prompt sheet
Prompt sheet 2