A. Laurent et al. (Eds.): IPMU 2014, Part II, CCIS 443, pp. 536–545, 2014. © Springer International Publishing Switzerland 2014 Probabilistic Solution of Zadeh’s Test Problems Boris Kovalerchuk Dept. of Computer Science, Central Washington University, 400 E. University Way, Ellensburg, WA 98926, USA borisk@cwu.edu Abstract. Zadeh posed several Computing with Words (CWW) test problems such as: “What is the probability that John is short?” These problems assume a given piece of information in the form of membership functions for linguistic terms including tall, short, young, middle-aged, and the probability density functions of age and height. This paper proposes a solution that interprets Zadeh’s solution for these problems as a solution in terms of probability spaces as defined in the probability theory. This paper also discusses methodological issues of relations between concepts of probability and fuzzy sets. 1 Introduction Zadeh’s test problems include: “What is the probability that Mary is middle-aged?”, “What is the probability that Mary is young?”, “What is the probability that John is short?”, and “What is the probability that John is tall?” [Zadeh, 2012, Belyakov et al, 2012]. These problems assume given information I in the form of membership functions μ tall , μ short , μ young , μ middle-aged , and the probability density functions of age P A and height P H Below we propose a solution that interprets Zadeh’s solution [2011] for these problems as a solution in terms of probability spaces as defined in the probability theory. First, we focus on one of the problems: “What is the probability that John is tall?” The given information I for this problem is: a specified membership function μ tall and the probability density function of Height(John), p H . Zadeh’s solutions for other listed problems are similar. In this notation, Zadeh defines the probability that John is tall as: du u p u H tall R ) ( ) ( μ  (1) where R is the real line, u belongs to R and u is the height value. Formula (1) is called the probability measure of the fuzzy set tall [Zadeh, 1968] and is considered as a translation of the given linguistic information I into a mathematical language with precisiation of meaning as Zadeh calls it. Similarly a formula for the probability that Mary is middle-aged is