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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
The probability distribution for \(x\) the number of adults who are against animal testing in a sample of two is \(P(X=0) = 0.49\), \(P(X=1) = 0.42\) and \(P(X=2) = 0.09\). A tree diagram can be used to visually represent this distribution.

Step by step solution

01

Defining the Random Variable

Define our random variable \(x\), which represents the number of adults in a sample of two who are against animal testing. Therefore, \(x\) can take on values 0, 1, and 2.
02

Calculate Probability for Each Value of \(x\)

Calculate the probability for each possible value of \(x\). The probability of getting \(0, 1,\) and \(2\) adults against animal testing for each possible value of \(x\) can be calculated as follows: - \(P(X=0)\): Probability of no adults (out of the two) oppose the animal testing. That means both support animal testing. It can be calculated as \(P(X=0) = (0.7) * (0.7) = 0.49\).- \(P(X=1)\): Probability of one adult (out of the two) oppose the testing. We have two such cases - first case is first adult oppose and the second support, second case is first support and the second oppose. So, \(P(X=1) = 2 * (0.3) * (0.7) = 0.42\).- \(P(X=2)\): Probability of both adults oppose the testing. It can be calculated as \(P(X=2) = (0.3) * (0.3) = 0.09\).
03

Construct the Probability Distribution

Create a table or list showing the possible values of \(x\) and the associated probability, as calculated above. We get the probability distribution as follows: - \(P(X=0) = 0.49\)- \(P(X=1) = 0.42\)- \(P(X=2) = 0.09\)
04

Draw the Tree Diagram

The tree diagram includes all possible outcomes in the sample of two adults. It's a visual representation of the calculation we made in step 2. It starts with two branches showing the outcomes for adult 1 (either they are against animal testing \(30\%\) or not \(70\%\)). Each of those branches splits into two again for the outcomes for adult 2. The ends of each branch list the combined outcome and the associated probability (which equals the product of probabilities along the path to that outcome).

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