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Chapter 9: Problem 66

a. What is the relationship between \(p=P(\text { success })\) and \(q=P(\text { failure }) ?\) Explain. b. Explain why the relationship between \(p\) and \(q\) can be expressed by the formula \(q=1-p\) c. If \(p=0.6,\) what is the value of \(q ?\) d. If the value of \(q^{\prime}=0.273,\) what is the value of \(p^{\prime} ?\)

Short Answer

Expert verified
Part a: \(p\) and \(q\) are the probabilities of success and failure respectively. They are complementary to each other, i.e., they add up to 1. Part b: The formula \(q=1-p\) comes from the fact that \(p\) and \(q\) are complementary. Part c: When \(p=0.6\), \(q=0.4\). Part d: When \(q=0.273\), \(p=0.727\).

Step by step solution

01

Understanding Probabilities

In any event, the sum of the probabilities of all possible outcomes is equal to 1. In a binary event, there are only two outcomes: success or failure. Therefore, the probability of success (\(p\)) plus the probability of failure (\(q\)) equals 1.
02

Relationship between Probabilities

Since \(p\) and \(q\) are the only two possibilities and they add up to 1, it can be said that \(q = 1 - p\), or alternatively, \(p = 1 - q\). This means that the probability of failure is 1 minus the probability of success, and vice versa.
03

Calculating \(q\) when \(p = 0.6\)

Using the derived relationship between \(p\) and \(q\) (i.e., \(q = 1 - p\)), when \(p = 0.6\), we can calculate \(q\) by substituting this value into the equation, which gives: \(q = 1 - 0.6\). Hence, \(q = 0.4\).
04

Calculating \(p\) when \(q = 0.273\)

Similarly, using the same equation (i.e., \(p = 1 - q\)), when \(q = 0.273\), we can calculate \(p\) by substituting this value into the equation, which gives: \(p = 1 - 0.273\). Hence, \(p = 0.727\).

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Key Concepts

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

Understanding Binary Outcomes
In probability theory, a binary outcome refers to a situation where there are only two possible results. For each event, you might be looking at successes and failures, wins and losses, heads and tails, or any other pair of opposing outcomes. This makes understanding binary outcomes quite important:
  • They simplify complex situations by reducing them to two possibilities.
  • They are widely used in areas such as decision-making, statistics, and machine learning.
By delving into binary outcomes, one can better grasp the nature of probabilities since they form the basis of many probability problems such as our original exercise.
Probability of Success
The probability of success, denoted as \( p \), is the likelihood that the desired event will occur. It's vital to understand this probability to make predictions and inform decisions:
  • It comes as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
  • In binary outcomes, knowing \( p \) helps in calculating the complement, \( q \), which is the probability of failure.
An excellent aspect of \( p \) is its ability to provide insights into how often you might expect successes over many trials, such as winning a coin toss multiple times.
Probability of Failure
Just as the probability of success has its own significance, so does the probability of failure, denoted as \( q \). This probability represents the chance of the non-occurrence of the event. When evaluating outcomes:
  • It plays a crucial role in assessing risk, especially in scenarios where failure has significant consequences.
  • Like \( p \), it ranges from 0 to 1, highlighting the range from impossible failure to certain failure.
In the context of our exercise, knowing the probability of failure lets us easily deduce the probability of success, as these two are complementary within binary systems.
Complementary Probabilities
Complementary probabilities explain the intrinsic relationship between the probability of success \( p \) and the probability of failure \( q \). They are two sides of the same coin in binary scenarios:
  • The sum of \( p \) and \( q \) equals 1. This means that if success is more likely, failure is less likely, and vice versa.
  • This complimentary relationship is algebraically expressed as \( q = 1 - p\) or \( p = 1 - q \).
Understanding complementary probabilities allows you to calculate one probability if you have the other, as demonstrated in the exercise application where values like \( p=0.6 \) make \( q=0.4 \), showing the seamless interplay between success and failure.

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Most popular questions from this chapter

You are testing the null hypothesis \(p=0.4\) and will reject this hypothesis if \(z \star\) is less than -2.05. a. If the null hypothesis is true and you observe \(z \star\) equal to \(-2.12,\) which of the following will you do? (1) Correctly fail to reject \(H_{o} .\) (2) Correctly reject \(H_{o} .(3)\) Commit a type I error. (4) Commit a type II error. b. What is the significance level for this test? c. What is the \(p\) -value for \(z \star=-2.12 ?\)

The full-time student body of a college is composed of \(50 \%\) males and \(50 \%\) females. Does a random sample of students \((30 \text { male, } 20\) female) from an introductory chemistry course show sufficient evidence to reject the hypothesis that the proportion of male and of female students who take this course is the same as that of the whole student body? Use \(\alpha=0.05\). a. Solve using the \(p\) -value approach. b. Solve using the classical approach.

State the null hypothesis, \(H_{o}\), and the alternative hypothesis, \(H_{a},\) that would be used to test these claims: a. The standard deviation has increased from its previous value of 24. b. The standard deviation is no larger than 0.5 oz. c. The standard deviation is not equal to \(10 .\) d. The variance is no less than \(18 .\) e. The variance is different from the value of \(0.025,\) the value called for in the specs.

It has been suggested that abnormal male children tend to be born to older- than-average parents. Case histories of 20 abnormal males were obtained, and the ages of the 20 mothers were as follows: $$\begin{array}{lllllllllll}31 & 21 & 29 & 28 & 34 & 45 & 21 & 41 & 27 & 31 \\\43 & 21 & 39 & 38 & 32 & 28 & 37 & 28 & 16 & 39\end{array}$$ The mean age at which mothers in the general population give birth is 28.0 years. a. Calculate the sample mean and standard deviation. b. Does the sample give sufficient evidence to support the claim that abnormal male children have olderthan-average mothers? Use \(\alpha=0.05 .\) Assume ages have a normal distribution.

It is important that the force required to extract a cork from a wine bottle not have a large standard deviation. Years of production and testing indicate that the no.9 corks in Applied Example 6.13 (p. 285 ) have an extraction force that is normally distributed with a standard deviation of 36 Newtons. Recent changes in the manufacturing process are thought to have reduced the standard deviation. a. What would be the problem with the standard deviation being relatively large? What would be the advantage of a smaller standard deviation? A sample of 20 randomly selected bottles is used for testing. Extraction Force in Newtons $$\begin{array}{llllllllll}\hline 296 & 338 & 341 & 261 & 250 & 347 & 336 & 297 & 279 & 297 \\\259 & 334 & 281 & 284 & 279 & 266 & 300 & 305 & 310 & 253 \\\\\hline\end{array}$$ b. Is the preceding sample sufficient to show that the standard deviation of extraction force is less than 36.0 Newtons, at the 0.02 level of significance? During a different testing, a sample of eight bottles is randomly selected and tested. Extraction Force in Newtons $$\begin{array}{rrrrrr}331.9 & 312.0 & 289.4 & 303.6 & 346.9 & 308.1 & 346.9 & 276.0\end{array}$$ c. Is the preceding sample sufficient to show that the standard deviation of extraction force is less than 36.0 Newtons, at the 0.02 level of significance? d. What effect did the two different sample sizes have on the calculated test statistic in parts b and c? What effect did they have on the \(p\) -value or critical value? Explain. e. What effect did the two different sample standard deviations have on the answers in parts b and c? What effect did they have on the \(p\) -value or critical value? Explain.
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