Question

The interactive system at Banker's Tryst can process 5 kinds of inquiries; the respective probabilities are $0.1,0.15,0.4,0.25$, and 0.1 . What is the probability the next 10 inquiries will include 1 of the first type, 2 of the second, 3 of the third, 3 of the fourth, and one of the fifth?

Name: Problem 9, Chapter 3: Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)
Uploaded: 2020-06-16T22:42:56-08:00
Duration: 59 s
Channel: Michael Dunne
Description: The interactive system at Banker's Tryst can process 5 kinds of inquiries; the respective probabilities are 0.1,0.15,0.4,0.25, and 0.1 . What is the probability the next 10 inquiries will include 1 of the first type, 2 of the second, 3 of the third, 3 of the fourth, and one of the fifth?

   The interactive system at Banker's Tryst can process 5 kinds of inquiries; the respective probabilities are $0.1,0.15,0.4,0.25$, and 0.1 . What is the probability the next 10 inquiries will include 1 of the first type, 2 of the second, 3 of the third, 3 of the fourth, and one of the fifth?

Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Arnold O. Allen 2nd Edition

Chapter 3, Problem 9

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Solved by Expert Michael Dunne

06/16/2020

Step 1

The problem describes multiple categories (types of inquiries) with specific probabilities for each category and asks for the probability of a particular combination of outcomes. This scenario fits a multinomial distribution, which generalizes the binomial Show more…

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The interactive system at Banker's Tryst can process 5 kinds of inquiries; the respective probabilities are $0.1,0.15,0.4,0.25$, and 0.1 . What is the probability the next 10 inquiries will include 1 of the first type, 2 of the second, 3 of the third, 3 of the fourth, and one of the fifth?

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Key Concepts

Multinomial Distribution

The multinomial distribution is an extension of the binomial distribution where each trial can result in one of several (more than two) outcomes. It provides a way to calculate the probability of achieving a specific combination of counts for different outcomes over a fixed number of independent trials, using the individual probabilities associated with each outcome.

Multinomial Coefficient

The multinomial coefficient, given by the formula n!/(n1!n2!...nk!) for a total of n trials with counts n1, n2, …, nk for each outcome, represents the number of distinct sequences in which the outcomes can occur. This combinatorial factor is essential in calculating probabilities under the multinomial distribution because it accounts for the different orderings of the outcomes that result in the same overall counts.

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