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Chapter 6: Problem 19

A mutual fund company offers its customers several different funds: a money market fund, three different bond funds, two stock funds, and a balanced fund. Among customers who own shares in just one fund, the percentages of customers in the different funds are as follows: \(\begin{array}{lr}\text { Money market } & 20 \% \\ \text { Short-term bond } & 15 \% \\ \text { Intermediate-term bond } & 10 \% \\ \text { Long-term bond } & 5 \% \\ \text { High-risk stock } & 18 \% \\ \text { Moderate-risk stock } & 25 \% \\ \text { Balanced fund } & 7 \%\end{array}\) A customer who owns shares in just one fund is to be selected at random. a. What is the probability that the selected individual owns shares in the balanced fund? b. What is the probability that the individual owns shares in a bond fund? c. What is the probability that the selected individual does not own shares in a stock fund?

Short Answer

Expert verified
a. The probability that the selected individual owns shares in the balanced fund is 0.07 or 7%. b. The probability that the individual owns shares in a bond fund is 0.30 or 30%. c. The probability that the selected individual does not own shares in a stock fund is 0.57 or 57%.

Step by step solution

01

Probability of Owning Shares in the Balanced Fund

The percentage of customers who own shares in the Balanced Fund is 7%. Therefore, the probability of a customer owning shares in the Balanced Fund is \(0.07\) or 7%.
02

Probability of Owning Shares in a Bond Fund

To find the probability that an individual owns shares in a bond fund, we sum up the percentages corresponding to each type of bond fund. There are three types of bond funds: Short Term Bond Fund(15%), Intermediate Term Bond Fund(10%), and Long Term Bond Fund(5%). Summing these up gives us \(15\% + 10\% + 5\% = 30\%\). Hence, the probability that a customer owns shares in a bond fund is \(0.30\) or 30%.
03

Probability of Not Owning Shares in a Stock Fund

To find the probability that the selected customer does not own shares in a stock fund, we subtract the combined percentages of the stock funds from 100. There are two types of stock funds: High-Risk Stock Fund (18%) and Moderate-Risk Stock Fund (25%). Hence, the percentage that owns shares in stock funds is \(18\% + 25\% = 43\%\). Substracting this from 100, we get \(100\% - 43\% = 57\% \). Therefore, the probability that a customer does not own shares in a stock fund is \(0.57\) or 57%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mutual Fund
A mutual fund is an investment vehicle made up of a pool of money collected from many investors to invest in securities such as stocks, bonds, money market instruments, and other assets. Mutual funds are operated by professional money managers, who allocate the fund's assets and attempt to produce capital gains or income for the fund's investors. One of the main advantages of mutual funds is that they provide an opportunity for individual investors to access professionally managed portfolios of equities, bonds, and other securities. The choice of investment, based on the fund type, comes with various levels of risk and return, catering to different investor goals.
Random Selection
In the context of probability, random selection refers to the process of choosing an individual in such a way that each individual of the population has an equal chance of being chosen. This is crucial in ensuring unbiased results. When a customer is randomly selected, it means the selection process does not favor any particular customer over another. Thus, the probability outcomes calculated from a random selection are reliable and reflective of the entire population's distribution. Random sampling helps in obtaining a representative sample, allowing conclusions to be drawn about the entire population.
Percentage Calculation
Percentage calculation is a fundamental concept in probability and statistics, often used to express probability as a percentage. To calculate the probability in percentage format, you multiply the probability by 100. For instance, if the probability of an event is 0.07, then in percentage terms it is 7%. This conversion helps in easily comparing and analyzing different probabilities. Often, problems will involve adding the probabilities of multiple related events, such as finding the total percentage of people owning shares in different types of bonds by summing the respective percentages.
Types of Funds
There are various types of funds available in the mutual fund industry, each catering to different investment styles and risk tolerances:
  • Money Market Fund: Generally considered low-risk, these funds invest in short-term debt securities and aim to provide returns with minimal risk.
  • Bond Funds: These include several types, such as short-term, intermediate-term, and long-term bond funds, which invest in bonds and seek to produce regular income.
  • Stock Funds: High-risk and moderate-risk stock funds invest in stocks with the aim of capital appreciation. They typically involve higher risks than money market or bond funds.
  • Balanced Fund: A blend of stocks, bonds, and other securities, balanced funds aim to reduce risk by diversifying across different types of investments.
Investors choose between these funds based on their financial goals, risk tolerance, and investment horizon. Each type of fund offers unique prospects and risks, requiring investors to carefully evaluate them before investing.

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Most popular questions from this chapter

Two different airlines have a flight from Los Angeles to New York that departs each weekday morning at a certain time. Let \(E\) denote the event that the first airline's flight is fully booked on a particular day, and let \(F\) denote the event that the second airline's flight is fully booked on that same day. Suppose that \(P(E)=.7, P(F)=.6\), and \(P(E \cap F)=.54\). a. Calculate \(P(E \mid F)\) the probability that the first airline's flight is fully booked given that the second airline's flight is fully booked. b. Calculate \(P(F \mid E)\).

A deck of 52 cards is mixed well, and 5 cards are dealt. a. It can be shown that (disregarding the order in which the cards are dealt) there are \(2,598,960\) possible hands, of which only 1287 are hands consisting entirely of spades. What is the probability that a hand will consist entirely of spades? What is the probability that a hand will consist entirely of a single suit? b. It can be shown that exactly 63,206 hands contain only spades and clubs, with both suits represented. What is the probability that a hand consists entirely of spades and clubs with both suits represented? c. Using the result of Part (b), what is the probability that a hand contains cards from exactly two suits?

Consider the following information about travelers on vacation: \(40 \%\) check work email, \(30 \%\) use a cell phone to stay connected to work, \(25 \%\) bring a laptop with them on vacation, \(23 \%\) both check work email and use a cell phone to stay connected, and \(51 \%\) neither check work email nor use a cell phone to stay connected nor bring a laptop. In addition \(88 \%\) of those who bring a laptop also check work email and \(70 \%\) of those who use a cell phone to stay connected also bring a laptop. With \(E=\) event that a traveler on vacation checks work email, \(C=\) event that a traveler on vacation uses a cell phone to stay connected, and \(L=\) event that a traveler on vacation brought a laptop, use the given information to determine the following probabilities. A Venn diagram may help. a. \(P(E)\) b. \(P(C)\) c. \(P(L)\) d. \(P(E\) and \(C)\) e. \(P\left(E^{C}\right.\) and \(C^{C}\) and \(L^{C}\) ) f. \(P(\) Eor C or \(L\) ) g. \(P(E \mid L)\) j. \(P(E\) and \(L)\) h. \(P(L \mid C)\) k. \(P(C\) and \(L)\) i. \(P(E\) and \(C\) and \(L)\) 1\. \(P(C \mid E\) and \(L)\)

A shipment of 5000 printed circuit boards contains 40 that are defective. Two boards will be chosen at random, without replacement. Consider the two events \(E_{1}=\) event that the first board selected is defective and \(E_{2}=\) event that the second board selected is defective. a. Are \(E_{1}\) and \(E_{2}\) dependent events? Explain in words. b. Let not \(E_{1}\) be the event that the first board selected is not defective (the event \(E_{1}^{C}\) ). What is \(P\left(\right.\) not \(\left.E_{1}\right)\) ? c. How do the two probabilities \(P\left(E_{2} \mid E_{1}\right)\) and \(P\left(E_{2} \mid\right.\) not \(\left.E_{1}\right)\) compare? d. Based on your answer to Part (c), would it be reasonable to view \(E_{1}\) and \(E_{2}\) as approximately independent?

A construction firm bids on two different contracts. Let \(E_{1}\) be the event that the bid on the first contract is successful, and define \(E_{2}\) analogously for the second contract. Suppose that \(P\left(E_{1}\right)=.4\) and \(P\left(E_{2}\right)=.2\) and that \(E_{1}\) and \(E_{2}\) are independent events. a. Calculate the probability that both bids are successful (the probability of the event \(E_{1}\) and \(E_{2}\) ). b. Calculate the probability that neither bid is successful (the probability of the event \(\left(\right.\) not \(\left.E_{1}\right)\) and \(\left(\right.\) not \(\left.E_{2}\right)\) ). c. What is the probability that the firm is successful in at least one of the two bids?
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