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Chapter 7: Problem 38

According to a study conducted in 2003 concerning the participation, by age, of \(401(\mathrm{k})\) investors, the following data were obtained: $$ \begin{array}{lccccc} \hline \text { Age } & 20 \mathrm{~s} & 30 \mathrm{~s} & 40 \mathrm{~s} & 50 \mathrm{~s} & 60 \mathrm{~s} \\ \hline \text { Percent } & 11 & 28 & 32 & 22 & 7 \\ \hline \end{array} $$ a. What is the probability that a \(401(\mathrm{k})\) investor selected at random is in his or her 20 s or 60 s? b. What is the probability that a \(401(\mathrm{k})\) investor selected at random is under the age of 50 ?

Short Answer

Expert verified
a. The probability that a 401(k) investor selected at random is in their 20s or 60s is \(\frac{18}{100}\) or 18%. b. The probability that a 401(k) investor selected at random is under the age of 50 is \(\frac{71}{100}\) or 71%.

Step by step solution

01

Calculate probabilities for each age group

First, we need to convert the given percentages for each age group into probabilities by dividing them by 100. For 20s: \(P(20s) = \frac{11}{100}\) For 30s: \(P(30s) = \frac{28}{100}\) For 40s: \(P(40s) = \frac{32}{100}\) For 50s: \(P(50s) = \frac{22}{100}\) For 60s: \(P(60s) = \frac{7}{100}\)
02

a. Probability of an investor in their 20s or 60s

To find the probability that a randomly selected 401(k) investor is in their 20s or 60s, we simply add the probabilities of these two age groups. \(P(20s \, or\, 60s) = P(20s) + P(60s) = \frac{11}{100} + \frac{7}{100} = \frac{18}{100}\) So, the probability that a 401(k) investor selected at random is in their 20s or 60s is \(\frac{18}{100}\) or 18%.
03

b. Probability of an investor under the age of 50

To find the probability that a randomly selected 401(k) investor is under the age of 50, we add the probabilities of the age groups 20s, 30s, and 40s. \(P(<50) = P(20s) + P(30s) + P(40s) = \frac{11}{100} + \frac{28}{100} + \frac{32}{100} = \frac{71}{100}\) So, the probability that a 401(k) investor selected at random is under the age of 50 is \(\frac{71}{100}\) or 71%.

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