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Chapter 4: Problem 33

A company that plans to hire one new employee has prepared a final list of six candidates, all of whom are equally qualified. Four of these six candidates are women. If the company decides to select at random one person out of these six candidates, what is the probability that this person will be a woman? What is the probability that this person will be a man? Do these two probabilities add up to \(1.0 ?\) If yes, why?

Short Answer

Expert verified
The probability of hiring a woman is \( \frac{2}{3} \), the probability of hiring a man is \( \frac{1}{3} \), and yes, these probabilities add to 1 because the events are complementary (either a man or a woman will definitely be hired).

Step by step solution

01

Calculate the total probability of hiring a woman

The probability of hiring a woman is calculated by dividing the number of favourable outcomes (number of women, which is 4) by the total number of outcomes (total number of candidates, which is 6). Formally, this probability \(P(W)\) can be written as: \(P(W) = \frac{{\text{{number of women}}}}{{\text{{total number of candidates}}}} = \frac{4}{6} = \frac{2}{3} \)
02

Calculate the total probability of hiring a man

Similarly, the probability of hiring a man can be calculated by dividing the number of favourable outcomes (number of men, which is 2) by the total number of outcomes (total number of candidates, which is 6). Formally, this probability \(P(M)\) can be written as: \(P(M) = \frac{{\text{{number of men}}}}{{\text{{total number of candidates}}}} = \frac{2}{6} = \frac{1}{3} \)
03

Ensure probabilities add up

To see if these two probabilities add up to 1, they are merely added together. Formally, this can be done as: \(P(M) + P(W) = \frac{1}{3} + \frac{2}{3} = 1 \). This result is to be expected, since either a man or a woman must definitely be hired.

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Key Concepts

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

Outcome
In probability, an outcome is a possible result that can occur from a given action or experiment. Each action or experiment is associated with different outcomes. For instance, if you flip a coin, the possible outcomes are heads or tails.
In the context of selecting a new employee from a list of candidates, every individual candidate represents a unique outcome. This means when the company makes the selection, each candidate being chosen stands as one possible outcome. Since the company has six candidates, there are six different outcomes when selecting one person at random.
Sample Space
The sample space is the set that includes all possible outcomes of a probability experiment. It's a way to organize and understand all the different results that can occur. Visualizing the sample space allows us to calculate probabilities more easily.
Using our hiring example, the sample space contains all names of the candidates who are equally likely to be hired. It accounts for both men and women on the list. If we list the candidates as A, B, C, D, E, and F, then the sample space can be represented as:
  • {A, B, C, D, E, F}
The sample space for this selection is always fixed, provided the number of candidates remains the same, regardless of gender proportions. It ensures that every possible outcome of the selection process is considered.
Favourable Outcomes
Favourable outcomes are those outcomes in the sample space that align with the event we are interested in. When calculating probabilities, identifying favourable outcomes is crucial.
In the scenario of hiring a woman, the favourable outcomes are the instances where a woman is selected. Since four out of the six candidates are women, there are four favourable outcomes if the event is selecting a woman.
Similarly, if we are interested in hiring a man, the favourable outcomes shift to the selection of a male candidate. With two men on the list, there exist two favourable outcomes in this case.
  • Probability of hiring a woman (P(W)) = Number of Women / Total Candidates = 4/6 = 2/3
  • Probability of hiring a man (P(M)) = Number of Men / Total Candidates = 2/6 = 1/3
To confirm that these cover the entire sample space, ensure both probabilities add up to 1, showing completeness of these favourable outcomes within the sample space. This understanding of favourable outcomes directly influences how we calculate and interpret probabilities.

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Most popular questions from this chapter

Given that \(A, B\), and \(C\) are three independent events, find their joint probability for the following. a. \(P(A)=.20, \quad P(B)=.46\), and \(P(C)=.25\) b. \(P(A)=.44, \quad P(B)=.27\), and \(P(C)=.43\)

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A screening test for a certain disease is prone to giving false positives or false negatives. If a patient being tested has the disease, the probability that the test indicates a (false) negative is .13. If the patient does not have the disease, the probability that the test indicates a (false) positive is .10. Assume that \(3 \%\) of the patients being tested actually have the disease. Suppose that one patient is chosen at random and tested. Find the probability that a. this patient has the disease and tests positive b. this patient does not have the disease and tests positive c. this patient tests positive d. this patient has the disease given that he or she tests positive

A statistical experiment has 10 equally likely outcomes that are denoted by \(1,2,3,4,5,6,7,8,9\), and 10 . Let event \(A=\\{3,4,6,9\\}\) and event \(B=\\{1,2,5\\}\). a. Are events \(A\) and \(B\) mutually exclusive events? b. Are events \(A\) and \(B\) independent events? c. What are the complements of events \(A\) and \(B\), respectively, and their probabilities?

A random sample of 400 college students was asked if college athletes should be paid. The following table gives a two-way classification of the responses. $$\begin{array}{lcc} \hline & \text { Should Be Paid } & \text { Should Not Be Paid } \\ \hline \text { Student athlete } & 90 & 10 \\ \text { Student nonathlete } & 210 & 90 \end{array}$$ a. If one student is randomly selected from these 400 students, find the probability that this student i. is in favor of paying college athletes ii. favors paying college athletes given that the student selected is a nonathlete iii. is an athlete and favors paying student athletes iv. is a nonathlete or is against paying student athletes b. Are the events "student athlete" and "should be paid" independent? Are they mutually exclusive? Explain why or why not.
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