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# Probabilistic reasoning and combinatorial analysis

DOI link for Probabilistic reasoning and combinatorial analysis

Probabilistic reasoning and combinatorial analysis book

# Probabilistic reasoning and combinatorial analysis

DOI link for Probabilistic reasoning and combinatorial analysis

Probabilistic reasoning and combinatorial analysis book

## ABSTRACT

If you are familiar with the probability calculus, you may solve the problem assuming that the prior probability of drawing any chip is 1/6. Given that the prior probability of drawing a red, round chip equals two times 1/6 (i.e., 1/3), and that the prior probability of drawing a red chip equals three times 1/6 (i.e., 1/2), you correctly conclude that if the chip that is drawn is red then the probability that it is round equals (1/3) / (1/2), that is 2/3. What if you are unfamiliar with the probability calculus (i.e., if you are in the same condition as almost all members of humankind)? We maintain that in this case too you will correctly solve the problem. The chip that is drawn is one of the three red chips. Since two of them are round, there are two possibilities out of three to obtain a round chip, that is, two chances out of three. Indeed, almost everybody arrives at this correct solution (see Gonzalez & Girotto, 2004), which is an example of an extensional evaluation of probability. Individuals reason extensionally about probabilities by considering and enumerating the various possibilities in which an event may occur, and then inferring the chances of its occurrence. For instance, reasoners consider the three possibilities in which they obtain a red chip, and the two possibilities in which it is round. They count each possibility as one chance, and correctly infer that the required solution is two chances out of three. Evidence exists that naive individuals do evaluate probabilities extensionally (see, e.g., Girotto & Gonzalez, 2001, 2002; Johnson-Laird, Legrenzi, Girotto, Sonino-Legrenzi, & Caverni, 1999; Legrenzi, Girotto, Sonino-Legrenzi, & Johnson-Laird, 2003; Sloman, Over, Slovak, & Stibel, 2003). However, the previous results concern problems, such as the six-chip one, in which the initial set of all possibilities (e.g., the set of

six chips) was given, and the required probability can be obtained by counting the possibilities in the appropriate sets. In many cases, however, elementary possibilities are not as readily ascertained as in the six-chip problem. Consider the following problem:

The dice problem We roll two regular dice and then we count the sum of the numbers that turn up. These sums can be: 2, 3, 4 . . . 11, 12. If the sum of the numbers turning up is odd, what is the probability that it is 7?