The surface of a circular dart board has a small center...

The surface of a circular dart board has a small center circle called the bull's-eye and 20 pie-shaped regions numbered from 1 to $20 .$ Each of the pie-shaped regions is further divided into three parts such that a person throwing a dart that lands on a specified number scores the value of the number, double the number, or triple the number, depending on which of the three parts the dart falls. If a person hits the bull's-eye with probability 0.01 . hits a double with probability 0.10 , hits a triple with probability $0.05,$ and misses the dart board with probability $0.02,$ what is the probability that 7 throws will result in no bull's-eyes, no triples, a double twice, and a complete miss once?

Key Concepts

Combinatorial Counting

Combinatorial counting involves determining the number of ways to arrange or select items before applying probability rules. This technique is used here to calculate how the various outcomes (such as doubles, misses, etc.) can be distributed among the seven throws. The selection process, often involving multinomial coefficients, is essential to account for all different sequences that yield the specified outcome counts.

Independent Events

In probability theory, independent events are events where the occurrence or non-occurrence of one event has no effect on the occurrence of any other. In the context of multiple dart throws, each throw is considered independent, meaning that the outcome of one throw does not influence the outcome of another. This principle is fundamental when calculating the joint probability of a sequence of events.

Multinomial Distribution

The multinomial distribution generalizes the binomial distribution to situations where each trial can result in more than two outcomes. It is used to calculate the probability of obtaining a specific combination of outcomes over multiple independent trials. In this problem, it allows us to determine the probability of having no bull's-eyes, no triples, exactly two doubles, one miss, and the remaining throws resulting in another outcome, by using the multinomial coefficient to count the number of ways this arrangement can occur.

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