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Chapter 5: Problem 27

Suppose that a satellite defense system is established in which four satellites acting independently have a 0.9 probability of detecting an incoming ballistic missile. What is the probability that at least one of the four satellites detects an incoming ballistic missile? Would you feel safe with such a system?

Short Answer

Expert verified
The probability that at least one satellite detects the missile is 0.9999. Yes, the system is highly reliable.

Step by step solution

01

- Understand the Probability of Detection Failure

Each satellite has a 0.9 probability of detecting the missile. Therefore, the probability that a single satellite fails to detect the missile is: \( P(\text{failure}) = 1 - 0.9 = 0.1 \)
02

- Calculate the Probability of All Satellites Failing

We need to find the probability that all four satellites fail to detect the missile. Since their actions are independent, multiply the failure probabilities of each satellite: \( P(\text{all fail}) = (0.1) \times (0.1) \times (0.1) \times (0.1) = 0.1^4 = 0.0001 \)
03

- Calculate the Probability of At Least One Satellite Detecting

The probability that at least one satellite detects the missile is the complement of all satellites failing to detect it. Thus, we use: \( P(\text{at least one detects}) = 1 - P(\text{all fail}) = 1 - 0.0001 = 0.9999 \)
04

- Conclusion on the Safety

Since the probability of at least one satellite detecting the missile is 0.9999, or 99.99%, the system is highly reliable. Yes, one would feel safe with such a system.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events
In probability theory, events are considered independent if the outcome of one event does not affect the outcome of another. In the context of the satellite defense system, when we say that the satellites are acting independently, we mean that the detection capability of one satellite has no bearing on the detection capability of another.
This independence is crucial for our probability calculations. Because the satellites act independently, the probability that all four satellites fail to detect the missile can be calculated by multiplying the probability of each satellite failing. Remember, for independent events:
  • P(A and B) = P(A) 脳 P(B)
  • P(A and B and C and D) = P(A) 脳 P(B) 脳 P(C) 脳 P(D)
Understanding this, we calculated the probability of all four satellites failing by raising the failure probability of one satellite (0.1) to the fourth power.
Complement Rule
The complement rule is a fundamental concept in probability. It states that the probability of an event occurring is equal to 1 minus the probability that it does not occur. This helps us find the probability of 'at least one' event happening.
In our exercise, we needed to find the probability that at least one satellite detects the missile. Calculating the probability for 'at least one' directly can be complicated, but using the complement rule simplifies this.
  • P(at least one detects) = 1 - P(none detect)
After calculating the probability that none of the satellites detect the missile (0.0001), we subtracted this value from 1 to find the probability that at least one detects the missile, resulting in 0.9999.
Probability Calculation
Probability calculation is the process of determining the likelihood of an event. It involves basic arithmetic and sometimes more advanced concepts like the complement rule and understanding independent events.
Let's take another look at our situation with the satellites:
  • Probability of one satellite failing: P(failure) = 1 - 0.9 = 0.1
  • Probability of all four satellites failing: P(all fail) = 0.1^4 = 0.0001
  • Probability of at least one satellite detecting the missile: P(at least one detects) = 1 - 0.0001 = 0.9999
This series of calculations shows the steps you need to follow when calculating probabilities in scenarios like this one. Regular practice with different exercises will help solidify your understanding of these concepts.

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Most popular questions from this chapter

The weather forecast says there is a \(10 \%\) chance of rain on Thursday. Jim wakes up on Thursday and sees overcast skies. Since it has rained for the past three days, he believes that the chance of rain is more likely \(60 \%\) or higher. What method of probability assignment did Jim use?

In 1991 , columnist Marilyn Vos Savant posted her reply to a reader's question. The question posed was in reference to one of the games played on the gameshow Let's Make a Deal hosted by Monty Hall. Her reply generated a tremendous amount of backlash, with many highly educated individuals angrily responding that she was clearly mistaken in her reasoning. (a) Using subjective probability, estimate the probability of winning if you switch. (b) Load the Let's Make a Deal applet at www.pearsonhighered.com/sullivanstats. Simulate the probability that you will win if you switch by going through the simulation at least 100 times. How does your simulated result compare to your answer to part (a)? (c) Research the Monty Hall Problem as well as the reply by Marilyn Vos Savant. How does the probability she gives compare to the two estimates you obtained? (d) Write a report detailing why Marilyn was correct. One approach is to use a random variable on a wheel similar to the one shown. On the wheel, the innermost ring indicates the door where the car is located, the middle ring indicates the door you selected, and the outer ring indicates the door(s) that Monty could show you. In the outer ring, green indicates you lose if you switch while purple indicates you win if you switch.

A flush in the card game of poker occurs if a player gets five cards that are all the same suit (clubs, diamonds, hearts, or spades). Answer the following questions to obtain the probability of being dealt a flush in five cards. (a) We initially concentrate on one suit, say clubs. There are 13 clubs in a deck. Compute \(P(\) five clubs \()=P(\) first card is clubs and second card is clubs and third card is clubs and fourth card is clubs and fifth card is clubs). (b) A flush can occur if we get five clubs or five diamonds or five hearts or five spades. Compute \(P\) (five clubs or five diamonds or five hearts or five spades). Note that the events are mutually exclusive.

How many different simple random samples of size 5 can be obtained from a population whose size is 50?

The following data represent the number of different communication activities (e.g., cell phone, text messaging, e-mail, Internet, and so on) used by a random sample of teenagers over the past 24 hours. $$ \begin{array}{lccccc} \text { Activities } & \mathbf{0} & \mathbf{1 - 2} & \mathbf{3 - 4} & \mathbf{5 +} & \text { Total } \\ \hline \text { Male } & 21 & 81 & 60 & 38 & \mathbf{2 0 0} \\ \hline \text { Female } & 21 & 52 & 56 & 71 & \mathbf{2 0 0} \\ \hline \text { Total } & \mathbf{4 2} & \mathbf{1 3 3} & \mathbf{1 1 6} & \mathbf{1 0 9} & \mathbf{4 0 0} \end{array} $$ (a) Are the events "male" and " 0 activities" independent? Justify your answer. (b) Are the events "female" and " \(5+\) activities" independent? Justify your answer. (c) Are the events \(" 1-2\) activities" and \(\cdot 3-4\) activities" mutually exclusive? Justify your answer. (d) Are the events "male" and "1-2 activities" mutually exclusive? Justify your answer.
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