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) to 5-star (highest). A sample of 25 mutual funds was selected from Morningstar Funds 500 (2008). The following counts were obtained:


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*Sixteen mutual funds were Domestic Equity funds.

*Thirteen mutual funds were rated 3-star or less.

*Seven of the Domestic Equity funds were rated 4-star.

*Two of the Domestic Equity funds were rated 5-star.


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?
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My original thought was to add the probability of the fund being a Domestic Equity fund (16/25) and the probability of being a fund with a 4-star or 5-star rating (9/25). This provides a 1. Then I subtracted (7/25) which is the number of funds that are Domestic Equity funds that are 4 or 5. This does not provide the expected answer of 19/25.
Will someone please explain how the book provides an answer of 19/25?
 A: In cases such as this it is best to work from first principles.
Denote by $A,B,C$ the following subsets of your probability space:
$A$ -- Domestic funds. $B$-- $4$ star rated funds, and $C$ -- $5$ star rated funds. Then you want the probability of the union $A\cup B\cup C$. One has:
$$P(A\cup B\cup C)=P(A)+P(B\cup C)-P(A\cap (B\cup C))$$
Now, $P(A)=\frac{16}{25}$ because there are $16$ domestic funds. The set $A\cap (B\cup C)$ is precisely the set of domestic funds rated $4$ or $5$ stars, so it has $9$ members, hence its probability is $\frac{9}{25}$. Since there are alltogether $12$ funds that are rated $4$ or $5$ funds, the probability of the middle term, $(B\cup C)$ is just $\frac{12}{25}$. As a result, the desired probability is:
$$\frac{16}{25}+\frac{12}{25}-\frac{9}{25}=\frac{19}{25}.$$
A: $16$ funds were Domestic Equity funds. So now we would like to find those funds that had 4 or 5 stars but weren't Domestic Equity Funds. We know that 13 mutual funds were rated 3-star or less. So $25-13 =12$ funds were rated 4 or 5 stars. But we don't want to add now Domestic Funds with this rating as they are already counted in the $16$ funds. So we have $12-7-2=3$. So the number of funds that we're interested in is $16+3=19$ and there are $25$ funds. So the probability we are looking for is $\frac{19}{25}$. 
