Assume that a single depth charge has a probability of (1)/(3) of sinking a submarine, (1)/(2) o

Name: Problem 5, Chapter 2: Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)
Uploaded: 2023-09-20T15:21:19-08:00
Duration: 1 min 17 s
Channel: Massimo Antonelli
Description: Assume that a single depth charge has a probability of (1)/(3) of sinking a submarine, (1)/(2) of damage, and (1)/(6) of missing. Assume also that two damaging explosions sink the sub. If four depth charges are dropped on a submarine, what is the probability that the sub sinks?

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Question

Assume that a single depth charge has a probability of $\frac{1}{3}$ of sinking a submarine, $\frac{1}{2}$ of damage, and $\frac{1}{6}$ of missing. Assume also that two damaging explosions sink the sub. If four depth charges are dropped on a submarine, what is the probability that the sub sinks?

   Assume that a single depth charge has a probability of $\frac{1}{3}$ of sinking a submarine, $\frac{1}{2}$ of damage, and $\frac{1}{6}$ of missing. Assume also that two damaging explosions sink the sub. If four depth charges are dropped on a submarine, what is the probability that the sub sinks?

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

Arnold O. Allen 2nd Edition

Chapter 2, Problem 5

Instant Answer

Solved by Expert Massimo Antonelli

09/20/2023

Step 1

- Let $S$ represent the event that a depth charge sinks the submarine. - Let $D$ represent the event that a depth charge damages the submarine. - Let $M$ represent the event that a depth charge misses the submarine. Show more…

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Assume that a single depth charge has a probability of $\frac{1}{3}$ of sinking a submarine, $\frac{1}{2}$ of damage, and $\frac{1}{6}$ of missing. Assume also that two damaging explosions sink the sub. If four depth charges are dropped on a submarine, what is the probability that the sub sinks?

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Key Concepts

Combinatorial Analysis

Combinatorial analysis involves counting the number of ways certain outcomes can occur over a series of independent events. This method is crucial for calculating the probability of compound events, such as the submarine sinking due to a direct hit or cumulative damages based on the counts of outcomes across multiple trials.

Multinomial Distribution

This concept extends the binomial distribution to situations where each independent trial has more than two possible outcomes, each with its own fixed probability. It is used to determine the probability of a specific combination of outcomes (such as sinking, damaging, or missing) occurring over several trials.

Independence of Events

Independence means that the outcome of one trial does not affect the outcome of any other trial. This assumption allows the probabilities associated with each depth charge to be multiplied together when determining the overall probability of combined outcomes.

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