Problem 38 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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 ?

Step-by-Step Solution

Answer

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 1: 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: \(P(60s) = \frac{7}{100}\)

Step 2: 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%.

Step 3: 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%.

Key Concepts

Age-Based Probability

When we talk about age-based probability, we refer to calculating the likelihood that an individual from a population falls within a certain age group. In the context of the 401(k) investment study, age categories such as 20s, 30s, 40s, etc., are used to define these groups.
Each age group has an associated percentage, which indicates the proportion of the overall population in that category. For example, if 11% of the investors are in their 20s, this directly reflects the probability of an investor being in their 20s when chosen at random.
To determine age-based probability:

• Identify the percentage provided for each age group.
• Convert this percentage into a probability by dividing by 100. For instance, for the 20s group, it would be \( \frac{11}{100} \).
• These probabilities can then be used to make decisions or interpret trends based on age groups in the population.

Percentages to Probability Conversion

Converting percentages to probabilities is a foundational skill in probability calculations. It allows us to shift from understanding data in a descriptive form to a statistical form.
Here's how to easily convert a percentage into a probability:

• Take the given percentage, which is often given in terms of 100.
• Divide this number by 100 to get a probability, which is a value between 0 and 1. For example, a 28% chance corresponds to a probability of \( \frac{28}{100} = 0.28 \).

Once you have the probabilities, these can be used for further calculations such as finding the likelihood of various events or scenarios. Probabilities can also be expressed in percentage form again for easier interpretation, showing how likely or unlikely an event is on a percentage basis.

Disjoint Events in Probability

Disjoint events, sometimes called mutually exclusive events, are events that cannot occur simultaneously. In probability, understanding disjoint events helps in accurate calculations.
In the case of the 401(k) example, when considering someone in their 20s vs. their 60s, these two events are disjoint. A person cannot simultaneously belong to both age groups, so the occurrence of one event precludes the occurrence of the other.
To find the probability of either event happening (investor in their 20s or 60s), you simply add their probabilities:

• Use the formula: \( P(A \text{ or } B) = P(A) + P(B) \), where A and B are disjoint events.
• For instance, we calculate \( P(20s \text{ or } 60s) = \frac{11}{100} + \frac{7}{100} = \frac{18}{100} \).

This principle helps when assessing any situation where two outcomes cannot occur together, giving a clearer understanding of the possible outcomes.