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In 1999, Sally Clark was convicted of the murder of her two sons. The data: In 1996, her first son died apparently of 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.

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What is the chance of a cot death to an affluent family, where neither parent smokes, and the mother is over 26? The data: The probability of a cot death in this family was estimated at 1 in 8500. So can you see where the figure of 1 in 73 million for two such deaths comes from?

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Is this an appropriate calculation to make? This calculation is only legitimate if the two events are independent. Do you think the second death is necessarily independent of the first? If it is not, why does this mean the calculation is not appropriate? If the second event depends on the first, then the probability that it occurs may well be quite different from that of the first – we certainly cant assume it remains the same.

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Another rare event. Two deaths in the same family from SIDS (Sudden Infant Death Syndrome) is rare. So too is a double murder of two babies. Professor Dawid estimates the probability of this as 1 in 2 billion.

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In 2003, Lucia de Berk was given a life sentence for the murder or attempted murder of patients in her care. In 2010, she was exonerated and received compensation and a public apology from the Dutch government. The data: An unexpected death of a baby at the hospital where she worked as a nurse in 2001 led to a number of unexpected deaths being re- examined. 9 deaths between September 2000 and September 2001 were deemed suspicious. Lucia de Berk had been on duty when all the deaths occurred. The hospital pressed charges against her.

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Why was she convicted? Why was she later exonerated? The data: A statistical calculation was used to prove that the probability of her shifts coinciding with the deaths of the patients was only 1 in 342 million, had the deaths occurred by chance. The court did not consider that the probability of 10 deaths by murder within a year in a hospital is also very unlikely. A re-trial in 2010 concluded that the deaths were either natural or caused by wrong treatment or bad management.

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How rare is this do you think? Do you think it ought to occur on average: to several families per year? to one family per year? only every few years? less often? How likely is a family in the UK to have 3 children all with the same birthday?

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Calculating the probability: Choose a date of birth for the first child. What is the probability that the second child is also born on this date? And the third? Now combine these probabilities to find the probability that all three children have the same birthday. (Assume that the dates of birth are independent of each other). How likely is a family in the UK to have 3 children all with the same birthday?

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Calculating the probability: How likely is a family in the UK to have 3 children all with the same birthday?

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A probability of 0.00075% or 7.5 chances in 1 million certainly makes this a rare event. But what havent we considered yet? The probability is for this event to occur to one family. So we also need to take into account how many families there are with at least 3 children. How likely is a family in the UK to have 3 children all with the same birthday?

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In the UK, there are 24 million households, of which 1 million are of a couple with 3 or more dependent children. Now how rare an event is this? A chance of 7.5 in 1 million for one family results in an expected 7.5 families per year with three children born on the same day. So it is a rare event for a particular family, but not for the nation as a whole! How likely is a family in the UK to have 3 children all with the same birthday?

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