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Chapter 5: Problem 18

According to a survey, \(30 \%\) of adults are against using animals for research. Assume that this result holds true for the current population of all adults. Let \(x\) be the number of adults who are against using animals for research in a random sample of two adults. Obtain the probability distribution of \(x\). Draw a tree diagram for this problem.

Short Answer

Expert verified
In a random sample of two adults, the probability that both are against using animals for research is 0.09, the probability that one is against and one is for using animals is 0.42, and the probability that both adults are for using animals for research is 0.49.

Step by step solution

01

Identify the parameters

Identify the number of trials (\(n\)) as the sample size, which is 2. Identify the probability of success in a single trial (\(p\)), which is 0.30. Thus, the variable \(x\) will hold the values 0, 1, 2 as the possible number of adults against using animals in research in a sample of 2 adults.
02

Calculate the Probability

Calculate the probability for the different values of \(x\): \(P(x = 0)\), \(P(x = 1)\) and \(P(x = 2)\) using the binomial probability formula: \(P(X = x) = C_{n}^{x} \cdot (p)^x \cdot (1-p)^{n-x}\) where \(C_{n}^{x}\) is the combination of \(n\) things taken \(x\) at a time. 1) For \(x = 0\), \(P(x = 0) = C_{2}^{0} \cdot (0.30)^0 \cdot (1 - 0.30)^{2 - 0} = 1 \cdot 1.00 \cdot 0.49 = 0.49\) 2) For \(x = 1\), \(P(x = 1) = C_{2}^{1} \cdot (0.30)^1 \cdot (1 - 0.30)^{2 - 1} = 2 \cdot 0.30 \cdot 0.70 = 0.42\) 3) For \(x = 2\), \(P(x = 2) = C_{2}^{2} \cdot (0.30)^2 \cdot (1 - 0.30)^{2 - 2} = 1 \cdot 0.09 \cdot 1.00 = 0.09\)
03

Draw the tree diagram

A tree diagram represents the events from a probability experiment. Each branch is labeled with a possible outcome and the probability of that outcome. For this particular problem, it would be more helpful to write out the solutions to the problem and describe the tree diagram, as it is difficult to represent a visual image in this format. The tree diagram for this problem would have 2 levels of branches. The first level corresponds to the outcome of the first pick (either For or Against). Each of these two branches would then further subdivide into two branches, also labeled For and Against, representing the outcome of the second pick.

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