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Chapter 8: Problem 3

In exercise, you are performing 5 independent Bernoulli trials with \(p=.1\) and \(q=.9 .\) Calculate the probability of each of the stated outcomes. Check your answer using technology. No successes

Short Answer

Expert verified
The probability of no successes in 5 independent Bernoulli trials with \(p=0.1\) and \(q=0.9\) is 0.59049.

Step by step solution

01

Understand the Bernoulli trial

A Bernoulli trial is a random experiment with exactly two possible outcomes: success, with probability \(p\), and failure, with probability \(q=1-p\). In this exercise, we have \(p=.1\) and \(q=.9\).
02

Use the binomial probability formula

Since we are asked to find the probability of no successes in 5 trials, we can use the binomial probability formula, which is given by: \[P(X=k) = \binom{n}{k}p^k q^{n-k}\] In our case, \(n=5\), \(k=0\) (no successes), \(p=.1\) and \(q=.9\).
03

Calculate the probability of no successes

Plugging the specific values of \(n\), \(k\), \(p\), and \(q\) into the formula, we get: \[P(X=0) = \binom{5}{0}(0.1)^{0}(0.9)^{5-0}\] \[P(X=0) = 1 (1)(0.9)^{5}\] Now, calculate \((0.9)^{5} \): \[(0.9)^{5}= 0.59049\] So, the probability of no successes is: \[P(X=0) = 1(1)(0.59049) = 0.59049\]
04

Check the answer using technology

To verify our answer, we can use software such as R, Python, or a calculator with a binomial probability function. Using R, we can calculate the probability with the following command: dbinom(0, 5, 0.1) This should return a value very close to our calculated probability of 0.59049, confirming our solution.

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Key Concepts

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

Binomial Probability Formula
Understanding the binomial probability formula is essential for solving problems involving a series of independent trials with two possible outcomes, like the ones found in Bernoulli trials. This formula is expressed as:
\[ P(X=k) = \binom{n}{k}p^k q^{n-k} \]
Here, \( P(X=k) \) denotes the probability of having exactly \( k \) successes in \( n \) independent trials. The symbol \( \binom{n}{k} \) is the binomial coefficient, equivalent to 'n choose k,' which calculates how many ways you can achieve \( k \) successes in \( n \) trials. The variable \( p \) represents the probability of success on a single trial, while \( q \) represents the probability of failure, which is \( 1-p \).

For example, to calculate the probability of having no successes (\( k=0 \)) in 5 trials (\( n=5 \)), with a success probability of 0.1 (\( p=0.1 \)), one would compute the binomial probability using the given values. Each part of the formula corresponds to an essential concept in understanding the statistical likelihood of an event: the number of combinations, the probability of repeated successes, and the probability of the remaining trials resulting in failures.
Independent Trials
The concept of independent trials is a cornerstone of probability theory. Independent trials mean that the outcome of any given trial does not affect the outcomes of subsequent trials. For a series of events to be considered independent, the probability of success must remain constant throughout the process. This is a critical assumption when using the binomial probability formula. In the context of Bernoulli trials, each event or trial should not influence the others鈥攌nowing the result of one does not give any information about the outcomes of others.

For students calculating binomial probabilities, ensuring the trials are independent simplifies the problem. If the trials were not independent, the probabilities could change after each event, complicating the calculation and making the straightforward binomial formula inapplicable. Checking for independence is an essential step before conducting probability calculations.
Probability Calculation
Probability calculation involves using the appropriate formula or method to determine the likelihood of a specific outcome. When it comes to calculating the probability of a set number of successes in Bernoulli trials, we use the binomial probability formula. After identifying the values of \( n \), \( k \), \( p \), and \( q \), we substitute them into the formula and perform the calculation. It鈥檚 beneficial to tackle each part of the formula step by step, by first determining the binomial coefficient, then raising the probabilities to their respective powers, and finally multiplying these results.

In the textbook example, the probability of no successes (\( k=0 \)) in 5 trials (\( n=5 \)) with success probability of 0.1 is calculated using the formula. This direct method of finding the likelihood of an event provides clarity and enables students to verify their answers using technology such as statistical software or graphing calculators, creating a comprehensive understanding of the principles governing probability calculation.

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Most popular questions from this chapter

Following is an excerpt from a full-page ad by MoveOn.org in the New York Times criticizing President G.W. Bush: \(^{33}\) On Tax Cuts: George Bush: "... Americans will keep, this year, an average of almost \(\$ 1,000\) more of their own money." The Truth: Nearly half of all taxpayers get less than \(\$ 100\). And \(31 \%\) of all taxpayers get nothing at all. The statements referred to as "The Truth" contradict the statement attributed to President Bush, right? Explain.

Shopping Malls A survey of all the shopping malls in your region yields the following probability distribution, where \(X\) is the number of movie theater screens in a selected mall: Compute the expected value \(\mu\) and the standard deviation \(\sigma\) of \(X\). (Round answers to two decimal places.) What percentage of malls have a number of movie theater screens within two standard deviations of \(\mu\) ?

Your scores for the 20 surprise math quizzes last semester were (out of 10 ) $$ \begin{aligned} &4.5,9.5,10.0,3.5,8.0,9.5,7.5,6.5,7.0,8.0 \\ &8.0,8.5,7.5,7.0,8.0,9.0,10.0,8.5,7.5,8.0 \end{aligned} $$ Use these raw data to construct a frequency table with the following brackets: \(2.1-4.0,4.1-6.0,6.1-8.0,8.1-10.0\), and find the probability distribution using the (rounded) midpoint values as the values of \(X\).

Compute the (sample) variance and standard deviation of the data samples given in Exercises \(1-8 .\) You calculated the means in the last exercise set. Round all answers to two decimal nlaces. $$ \frac{1}{2}, \frac{3}{2},-4, \frac{5}{4} $$

I \(\vee\) Supermarkets A survey of supermarkets in the United States yielded the following relative frequency table, where \(X\) is the number of checkout lanes at a randomly chosen supermarket: \(^{49}\) \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \(\boldsymbol{x}\) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \(\boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x})\) & \(.01\) & \(.04\) & \(.04\) & \(.08\) & \(.10\) & \(.15\) & \(.25\) & \(.20\) & \(.08\) & \(.05\) \\ \hline \end{tabular} a. Compute the mean, variance, and standard deviation (accurate to one decimal place). b. As financial planning manager at Express Lane Mart, you wish to install a number of checkout lanes that is in the range of at least \(75 \%\) of all supermarkets. What is this range according to Chebyshev's inequality? What is the least number of checkout lanes you should install so as to fall within this range?
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