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Chapter 5: Problem 20

In a group of 12 persons, 3 are left-handed. Suppose that 2 persons are randomly selected from this group. Let \(x\) denote the number of left-handed persons in this sample. Write the probability distribution of \(x\). You may draw a tree diagram and use it to write the probability distribution. (Hint: Note that the selections are made without replacement from a small population. Hence, the probabilities of outcomes do not remain constant for each selection.)

Short Answer

Expert verified
The probability distribution of \(x\) (number of left-handed persons) is: \(P(x=0) = 27/44\), \(P(x=1) = 15/44\), \(P(x=2) = 1/22\).

Step by step solution

01

Understand The Exercise

From the exercise, there are 12 persons in the group with 3 of them being left-handed. Two persons are to be selected from the group and this means two events are to be considered. The number of left-handed persons, denoted by \(x\), can be either 0, 1, or 2 as we are only drawing two people. We need to find the probabilities for each of these outcomes.
02

Draw the Probability Tree

The tree diagram will consist of two stages (selections) with branches emanating from each stage. The branches represent the outcomes of selecting a left-handed person (L) or a right-handed person (R). Every branch in the tree diagram will represent a possible outcome, and the probability of each outcome will be calculated by multiplying the probabilities along the branches.
03

Calculate the Probabilities

The probability of drawing a left-handed person in the first draw is \(P(L1) = 3/12 = 1/4\). After removing one left-handed person, the probability of doing so on the second draw is \(P(L2|L1) = 2/11\). Multiply these probabilities for the outcome of drawing two left-handed persons \(P(L1 and L2) = P(L1) \times P(L2|L1) = 1/4 \times 2/11 = 1/22\). Do this similarly for all other outcomes (L1 and R2, R1 and L2, and R1 and R2), noting that \(P(R1) = 1 - P(L1) = 9/12 = 3/4\), and \(P(R2|R1) = 1 - P(L2|R1) = 8/11\) and \(P(R2|L1) = 1- P(L2|L1) = 9/11\).
04

Write the Probability Distribution

After calculating the probabilities, construct the probability distribution which shows all possible values of \(x\) (0, 1, 2) and their respective probabilities. \(P(x=0) = P(R1 and R2)\), \(P(x=1) = P(L1 and R2) + P(R1 and L2)\), \(P(x=2) = P(L1 and L2)\). Fill in the probabilities you calculated in Step 3.

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Most popular questions from this chapter

Which of the following are binomial experiments? Explain why. a. Drawing 3 balls with replacement from a box that contains 10 balls, 6 of which are red and 4 are blue, and observing the colors of the drawn balls b. Drawing 3 balls without replacement from a box that contains 10 balls, 6 of which are red and 4 are blue, and observing the colors of the drawn balls c. Selecting a few households from New York City and observing whether or not they own stocks when it is known that \(28 \%\) of all households in New York City own stocks

Let \(N=8, r=3\), and \(n=4\). Using the hypergeometric probability distribution formula, find a. \(P(x=2)\) b. \(P(x=0)\) c. \(P(x \leq 1)\)

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The binomial probability distribution is symmetric for \(p=.50\), skewed to the right for \(p<.50\), and skewed to the left for \(p>.50\). Illustrate each of these three cases by writing a probability distribution table and drawing a graph. Choose any values of \(n\) (equal to 4 or higher) and \(p\) and use the table of binomial probabilities (Table I of Appendix \(\mathrm{C}\) ) to write the probability distribution tables.

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