Information about mutual funds provided by Morningstar...

Information about mutual funds provided by Morningstar Investment Research includes the type of mutual fund (Domestic Equity, International Equity, or Fixed Income) and the Morningstar rating for the fund. The rating is expressed from 1-star (lowest rating) to 5-star (highest rating). A sample of 25 mutual funds was selected from Morningstar Funds500 (2008) . The following counts were obtained: $\bullet$ Sixteen mutual funds were Domestic Equity funds. $\bullet$ Thirteen mutual funds were rated 3 -star or less. $\bullet$ Seven of the Domestic Equity funds were rated 4-star. $\bullet$ Two of the Domestic Equity funds were rated 5 -star. Assume that one of these 25 mutual funds will be randomly selected in order to learn more about the mutual fund and its investment strategy. a. What is the probability of selecting a Domestic Equity fund? b. What is the probability of selecting a fund with a 4 -star or 5 -star rating? c. What is the probability of selecting a fund that is both a Domestic Equity fund and a fund with a 4 -star or 5 -star rating? d. What is the probability of selecting a fund that is a Domestic Equity fund or a fund with a 4 -star or 5 -star rating?

Information about mutual funds provided by Morningstar Investment Research includes the type of mutual fund (Domestic Equity, International Equity, or Fixed Income) and the Morningstar rating for the fund. The rating is expressed from 1-star (lowest rating) to 5-star (highest rating). A sample of 25 mutual funds was selected from Morningstar Funds500 (2008) . The following counts were obtained:
$\bullet$ Sixteen mutual funds were Domestic Equity funds.
$\bullet$ Thirteen mutual funds were rated 3 -star or less.
$\bullet$ Seven of the Domestic Equity funds were rated 4-star.
$\bullet$ Two of the Domestic Equity funds were rated 5 -star.
Assume that one of these 25 mutual funds will be randomly selected in order to learn more about the mutual fund and its investment strategy.
a. What is the probability of selecting a Domestic Equity fund?
b. What is the probability of selecting a fund with a 4 -star or 5 -star rating?
c. What is the probability of selecting a fund that is both a Domestic Equity fund and a fund with a 4 -star or 5 -star rating?
d. What is the probability of selecting a fund that is a Domestic Equity fund or a fund with a 4 -star or 5 -star rating?

Key Concepts

Addition Rule for Probabilities

The addition rule is a principle used to compute the probability of the union of two events. It states that the probability of either event A or event B occurring is equal to the sum of their individual probabilities minus the probability of both events occurring together. This rule prevents double-counting when events are not mutually exclusive.

Set Operations: Union and Intersection

Set operations in probability, such as union and intersection, help in analyzing events that may overlap or occur simultaneously. The union of two events represents outcomes in either event, whereas the intersection represents outcomes where both events occur. These operations are pivotal in accurately calculating probabilities without overcounting shared outcomes.

Basic Probability

Basic probability involves determining the likelihood of an event by dividing the number of favorable outcomes by the total number of outcomes in a sample space. It forms the foundation of probability theory by providing a straightforward quantitative measure of uncertainty in various scenarios.

Sample Space

The sample space is the complete set of all possible outcomes of an experiment or random process. In probability problems, it serves as the reference set from which the likelihood of events is assessed, ensuring that probability calculations consider every possibility.

Transcript

00:01 All right, so we're told that a question and a question that advisor, let's say, has 25 mutual funds rated from one star to five star.

00:07 Sixteen of these mutual funds were designated as domestic equity funds.

00:11 13 were rated three star or less total out of the 25.

00:15 Seven of the domestic mutual funds are rated four star and two of the domestic mutual funds are rated five star.

00:20 And so basically we're giving four parts of the question and we define probabilities for the different four parts.

00:25 So first we're going to look at the probability of picking a domestic equity fund.

00:29 And so all we're going to do for this is take our total of domestic equity funds and divided by our total funds altogether.

00:36 And so we have 16 total, we have 16 total domestic funds and then a 25 total of all of our funds.

00:43 And so when we just divide 16 divided by four, i mean 16 divided by 25, sorry.

00:48 Also, you can do 16 times four to get your percentage and put a decimal place in front or front.

00:53 This is going to be 0 .64 as our probability.

00:58 Then i ask for the probability of having a four -star or a five -star rating of a mutual fund.

01:04 And so basically, we're actually giving the complement to this probability, and we're going to have to do a little bit of computation to figure this out because it's not directly given to us.

01:12 And so we're going to use our information of 13 rated 3 -star or less.

01:16 And so if we know that they're rated from 1 star to 5 -star and 13 -rated 3 -ar -less, the only other options that aren't accounted for are 4 -star and 5 -star.

01:25 And all we have to do is subtract our total 25 from 13 to realize that the other 12 were the other 12 have to be four star or five star.

01:36 So this is going to be 12 over 25.

01:39 And once again, we have something over 25.

01:41 All you have to do is multiply your numerator to divide it by four and put a decimal in the front.

01:45 So this is going to be 0 .48 as our probability for here.

01:50 All right.

01:51 So for c, we're asked to find the probability that a mutual fund is a domestic equity fund as well.

01:56 As and four or five star.

01:59 And so we're literally given this information in the last two bullet points, seven of the domestic mutual funds were rated four star and two of the domestic mutual funds were rated five star...

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