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Chapter 4: Problem 8

Find the indicated complements. When the author observed a sobriety checkpoint conducted by the Dutchess County Sheriff Department, he saw that 676 drivers were screened and 6 were arrested for driving while intoxicated. Based on those results, we can estimate that \(P(I)=0.00888\), where \(I\) denotes the event of screening a driver and getting someone who is intoxicated. What does \(P(\bar{I})\) denote, and what is its value?

Short Answer

Expert verified
P(\bar{I}) denotes screening a driver and not finding someone who is intoxicated, and its value is 0.99112.

Step by step solution

01

- Understand the notation and given data

The probability of screening a driver and finding someone who is intoxicated is given as \( P(I) = 0.00888 \). This means that 0.888% of the drivers screened were found intoxicated.
02

- Define the complement event

\( \bar{I} \) denotes the complement of the event \( I \). The complement of \( I \) is screening a driver and NOT finding someone who is intoxicated.
03

- Apply the Complement Rule

According to the Complement Rule in probability, \( P(\bar{I}) = 1 - P(I) \). This rule states that the probability of the complement event is equal to 1 minus the probability of the event.
04

- Compute \( P(\bar{I}) \)

Calculate the value of \( P(\bar{I}) \): \[ P(\bar{I}) = 1 - 0.00888 = 0.99112 \]

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Key Concepts

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

complement rule
Let's start by understanding the complement rule in probability. Probability complements deal with the occurrence of events and their opposite scenarios. For any event, the probability of the event happening plus the probability of it not happening equals 1.
To express this mathematically:
  • For an event, say, I, the probability is denoted by P(I).
  • The complement of event I, meaning the event does not happen, is written as \( \bar{I} \).
The Complement Rule states:
\[ P(\bar{I}) = 1 - P(I) \]
So, if you know the probability of an event happening, you can easily find the probability of it not happening using this rule.
probability calculation
Now, let's discuss probability calculation using the provided data. According to the example, the probability of screening a driver and finding them intoxicated was given as P(I) = 0.00888.
This value indicates that approximately 0.888% of the drivers screened were found intoxicated.
Using the complement rule, calculate the probability of not finding an intoxicated driver:
  • First, identify P(\bar{I}), which denotes the probability of not finding an intoxicated driver.
  • Then, apply the complement rule formula.

Substituting the given value:
\[ P(\bar{I}) = 1 - P(I) = 1 - 0.00888 = 0.99112 \]
So, the probability of screening a driver and not finding them intoxicated is approximately 0.99112, or 99.112%.
event notation
Understanding event notation is crucial in statistical probability. In this example, event I represents the specific occurrence of screening a driver and finding them intoxicated. Its probability notation is P(I).
The complement of this event, written as \(\bar{I}\), represents the occurrence of screening a driver and not finding them intoxicated. Both of these notations help in clearly stating and solving probability problems.

Here鈥檚 a quick summary of important points:
  • P(I) is the probability of the event I happening.
  • P(\bar{I}) is the probability of the event I not happening (its complement).
Proper use of notation simplifies understanding complex problems and ensures accurate results.

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Most popular questions from this chapter

Let \(A=\) the event of getting at least 1 defective iPhone when 3 iPhones are randomly selected with replacement from a batch. If \(5 \%\) of the iPhones in a batch are defective and the other \(95 \%\) are all good, which of the following are correct? a. \(P(\bar{A})=(0.95)(0.95)(0.95)=0.857\) b. \(P(A)=1-(0.95)(0.95)(0.95)=0.143\) c. \(P(A)=(0.05)(0.05)(0.05)=0.000125\)

Find the probability and answer the questions. When Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 428 green peas and 152 yellow peas. Based on those results, estimate the probability of getting an offspring pea that is green. Is the result reasonably close to the expected value of \(3 / 4\), as Mendel claimed?

Express all probabilities as fractions. In the game of blackjack played with one deck, a player is initially dealt 2 different cards from the 52 different cards in the deck. A winning "blackjack" hand is won by getting 1 of the 4 aces and 1 of 16 other cards worth 10 points. The two cards can be in any order. Find the probability of being dealt a blackjack hand. What approximate percentage of hands are winning blackjack hands?

Answer the given questions that involve odds. In the Kentucky Pick 4 lottery, you can place a "straight" bet of \(\$ 1\) by selecting the exact order of four digits between 0 and 9 inclusive (with repetition allowed), so the probability of winning is \(1 / 10,000\). If the same four numbers are drawn in the same order, you collect \(\$ 5000\), so your net profit is \(\$ 4999\). a. Find the actual odds against winning. b. Find the payoff odds. c. The website www.kylottery.com indicates odds of \(1: 10,000\) for this bet. Is that description accurate?

Find the probability. It has been reported that \(20 \%\) of iPhones manufactured by Foxconn for a product launch did not meet Apple's quality standards. An engineer needs at least one defective iPhone so she can try to identify the problem(s). If she randomly selects 15 iPhones from a very large batch, what is the probability that she will get at least 1 that is defective? Is that probability high enough so that she can be reasonably sure of getting a defect for her work?
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