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Probability

# Thinking Probabilistically

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which of the following scenarios is more probable?

The world is probabilistic. Even "nearly impossible" things, like winning the lottery, have some calculable probability of occurring.

Probability gives a nearly universal framework for analyzing the probabilistic world around us, with applications that stretch from games to sports to finance to engineering to medicine. For example, as later problems will show, probability can help us determine our confidence that someone committed a crime given some evidence in the case.

Whose statement is the most reasonable?

Ali: I just flipped 3 heads in a row with a fair coin. My next flip is very likely to also be heads.

Ben: I rolled a fair six-sided die 10 times and I never rolled a 6. The next roll is especially likely to be a 6 because I am "due" for one.

Cam: Usually, if it rains in Brilliantia (40 km west of where I live), it rains here a couple of hours later. It just started raining in Brilliantia, so it will probably rain here soon.

Probability allows for the quantification of extremely rare events. For example, suppose Tim has two ways to try to win a large sum of money:

• Win at a lottery with a chance of 1 in 300 million.

• Roll a fair six-sided die 20 times and roll all 6s.

Which winning event is more likely to happen?

Which of these events is the most likely to happen when flipping a fair coin?

• Flip 2 or more heads when flipping 3 coins.

• Flip 20 or more heads when flipping 30 coins.

• Flip 200 or more heads when flipping 300 coins.

Instead of making an explicit calculation, think about how "surprising" each of these outcomes would be.

Probability can help avoid logical fallacies, quantify rare events, and compare the likelihood of various outcomes. However, like any framework, it only works if the underlying assumptions are reasonable.

An incident of the misuse of probability occurred in 1999, when Sir Roy Meadow successfully convicted Sally Clark of double homicide of her two children. The first died in 1996, aged 10 weeks; the second died in 1998, aged 8 weeks. Sally Clark claimed that both of her children died of Sudden Infant Death Syndrome (SIDS), a rare condition in which an infant suddenly and inexplicably dies.

The final two problems explore Sir Roy Meadow's probabilistic argument against Sally Clark.

Sir Roy Meadow argued that in lower-risk families (e.g., a non-smoking household) like the Clark's, the probability of a single SIDS death was about 1 in 8543. He further argued that the events of two children dying of SIDS in a single family are independent, so to find the probability of two SIDS deaths you could square the probability of a single SIDS death.

What approximate probability did Sir Roy Meadow claim for the probability that both of Sally Clark's infants would have died from SIDS?

Sir Roy Meadow argued that his probability calculations implied that it was nearly certain that Sally Clark murdered both of her children. Which of the following arguments refutes this?

$$\boxed{1}$$ It is not clear that SIDS (which could be caused by some genetic predisposition) is actually independent across children from the same parents.

$$\boxed{2}$$ The "a priori" probability that Sally Clark would commit a double homicide was also incredibly rare, if not more rare than double-SIDS.

Note: "A priori" refers to the probability that Sally Clark would commit such a crime before taking into account the information of the deaths, which is the same type of probability that Sir Roy Meadow calculated. You can assume the facts are true, and you are simply evaluating reasonableness of the arguments.

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