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One approach to de Méré’s problem is to indeed imagine, like solution 3 in the previous section, the two gentlemen finishing their series of seven games many times over many days and determining the count of days one would expect each player to take the $200 pot. The proportion of wining days for each suggests the ratio to divide the pot.

This idea relies on a philosophical belief:

If a certain outcome of an experiment has a probability \(p \%\) of occurring, then in performing that experiment many, many times, we’d expect to see that outcome about  \(p \% \) of the time.

Without defining the term “probability” or “chance,” we intuitively do feel that actions and experiments with varying outcomes have certain inherent values associated with them—called probabilities—and that these values become manifest when running the same action or experiment over and over again a very large number of times.

For example, if we roll a die a million times, we feel we’ll likely see a roll of 5 about one-sixth of the time. If we toss a coin a 90,000 times, we feel we’ll see about 45,000 of the tosses land heads. And so on.

We also feel that this vague intuitive idea works in reverse. For example, if we toss a coin 500 times and it lands heads for a count of 403 of those tosses, then we’ll all strongly suspect that the coin is biased (and biased with a “probability” of about 80% for tossing a head).

Even with just this vague understanding of matters we can nut our way through some challenging probability problems by imagining repeating an experiment a large number times.

Example: A bag contains 8 Tuscan sunset orange balls and 2 Tahitian sunrise orange balls. The color difference is very subtle and only 70% of people can correctly identify the color of a ball when handed one. Lulu pulls a ball out of the bag at random and tells you over the phone that she pulled out a Tahitian sunrise ball. What are the chances that the ball she holds in her hand really is Tahitian sunrise?

Answer: Let’s assume, as the statistic given suggests, that there is a 70% chance that Lulu can correctly identify the color of a ball when handed one.

Now imagine Lulu conducting this ball-picking experiment a large number of times, say 100 times (and that her chances of correctly identifying colors does not change.)

About 80 of the balls Lulu pulls out will be Tuscan sunset.  Of those, she’ll identify about \(0.7 \times 80 = 56\) of them correctly as Tuscan sunset and 24 she’ll incorrectly say are Tahitian sunrise. About 20 of the balls Lulu pulls out will be Tahitian sunrise, of which \(0.7 \times 20 = 14\) she’ll correctly identify as such. However, she’ll call  of them Tuscan sunset.

Thus in these 100 runs of the experiment we see that Lulu will say “Tahitian sunrise” about \(24+14=38\) times and will be correct in saying this 14 of those times. This shows that the probability of her ball really being Tahitian sunrise is\(\dfrac{14}{38} \approx 37\%\), pretty low!

Comment: Even though eye-witnesses can identify suspects with reasonably high certainty, the probabilities that they are actually correct in their claims can still be low.

Exercise: Suppose that one-percent of the population has a certain disease. A test for the disease will return a positive result for  99% of the people with the disease and give a false negative for 1% of those with the disease. The test will correctly give a negative result for 95% of those people without the disease and otherwise gives a false positive.

You have just tested positive for the disease. Show that there is a one-in-six chance that you actually have the disease.

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