### Cyclic Quad Jigsaw

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

### Angle Trisection

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

### Beelines

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

# Probability in Court

##### Age 14 to 18 Challenge Level:

This resource is part of the collection Probability and Evidence.

Take a look at this video, where Prof Dawid discusses some of the ways in which probability has been used inappropriately in court cases.

### Sally Clark and Sudden Infant Death Syndrome (SIDS)

• In 1996, her first son died apparently of SIDS (also known as "cot death") at a few weeks of age.
• In 1998, her second son died similarly.
• She was convicted for their murder when the expert witness for the prosecution said the chance of the two deaths happening accidentally was 1 in 73 million.
• There was virtually no other evidence

The expert used the following method to calculate the probability:

• The probability of SIDS in an affluent family where neither parent smokes and the mother is aged over 26 is approximately $\tfrac{1}{8500}$.
• The probability of this happening to both of the children is therefore $\left( \tfrac{1}{8500} \right)^2 = \tfrac{1}{72,250,000}$.

What assumption has the expert made?
When you've thought about this question, click the button below.

The expert has assumed that the events are independent: so the second death is no more or less likely to occur once you know about the first one.

Do you think that this is a reasonable assumption to make?
It might help you to know that:
• SIDS is the name given to sudden, unexpected and unexplained deaths in apparently healthy babies.
• There may be as yet undiscovered genetic causes for SIDS.
• There may be hidden environmental causes for SIDS.

Two cases of SIDS in the same family is an extremely rare event.
So too is a double murder of two babies - this probability can be estimated as 1 in 2 billion.

Does this affect the probability that Sally Clark murdered her children?

It might help to consider what would happen in 2 billion random families.
How many of these would have a double murder of children?
In how many of these would two children suffer from SIDS?

In what proportion of families where two children have died were the children murdered?

Sally Clark served three years in prison, before having her convictions overturned.

### Lucia de Berk and Hospital Murders

Lucia de Berk, a nurse from the Netherlands, was given a life sentence for the murder or attempted murder of patients in her care in 2003.

Between September 2000 and September 2001, there were 9 unusual deaths in the hospital, which all occurred when Lucia de Berk was on duty.

A statistical calculation was used to "prove" that the probability of her shift coinciding with the deaths of the patients by chance had a probability of $\tfrac{1}{342,000,000}$.

Does this mean that the probability of Lucia being innocent was $\tfrac{1}{342,000,000}$? What else should a jury consider?

See how your thoughts compare to the analysis below:

The court needs to weigh up two different explanations: murder or coincidence. The argument that the deaths were unlikely to have occurred by chance ... is not that meaningful on its own ... What matters is the relative likelihood of the two explanations. However, the court was given an estimate for only the first scenario.
Mark Buchanan, Statistics: conviction by numbers (Nature 2007)

Lucia de Berk was eventually released, and the Dutch government apologised, in 2010.