By John G Michaels; Kenneth H Rosen

Each one bankruptcy of this supplement to any path in discrete arithmetic examines an program to enterprise, machine technological know-how, the sciences, or the social sciences. scholars paintings those chapter-length versions utilizing uncomplicated suggestions of combinatorics, graphs, recursion, family members, good judgment, likelihood, and finite nation machines

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0 Example 9 With reference to Example 4, what is the probability that, having planted one of the seeds in the package (chosen at random), we observe the sequence of flower colors "red, red, yellow, red" over the first four years? 3 . 3 . 5 . 018 . o 28 Applications of Discrete Mathematics Example 10 Referring again to Example 4, compute the probability that after having planted a randomly chosen seed, we must wait 3 years for the first red flower. Solution: This means that one of the following sequences of outcomes must have occurred: 828281 828381 838281 83 8 3 8 1.

We will show that knowing the transition probabilities Pi; and the initial probability distribution Q of a Markov chain suffices for determining all probabilities of interest in connection with the Markov chain. Indeed, all such probabilities can be computed if we know the probability of observing any specific sequence of outcomes; any other events of interest are made up of such sequences. , having 1 dollar left after flipping the coin twice. Solution: We must compute p(Xo = 3, Xl of conditional probability, we have = = 2, X 2 = 1).

Example 1 A weak order may be represented in the following manner: {C>- B,A >- E>- D, F}, where the candidates to the left of >- are preferred to those to the right, and the voter is indifferent between candidates separated by commas. Thus, in this example, the voter prefers candidate C to all alternatives; is indifferent between B and A which are preferred to D, E, and F; and is also indifferent between D and F. Every possible preference of a voter may be represented in the above manner. 0 Definition 3 Let W be the set of all weak orders on n candidates and assume there are N voters.