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Question 635788: A survey of 1000 Americans was taken to analyze their investments. Of those surveyed, 650 had invested in stocks, 550 in bonds, and 400 in both stocks and bonds. Of those surveyed,
a. How many invested in only stocks?
b. How many invested in stocks or bonds?
c. How many did not invest in either stocks or bonds?
Thank you sooo much
Answer by DrBeeee(684)   (Show Source):
You can put this solution on YOUR website! One good way to solve this problem is to use (or visualize) a Venn diagram.
First we have a large oval that contains the total population of your sample space, 1000 in this case. Now place/visualize two non-intersecting circles within the large oval. One circle contains the total number of stock investors, 650 in this case. The other circle contains the total number of bond investors, 550 in this case. Remember that our circles do not overlap inside the oval of 1000 people. But this (non-overlap) is not possible because 650 plus 550 is 1200.
Now we merely move the circles so that they overlap. How many people are in the overlapped area? Well it is the number of people who invest in both stocks and bonds, 400 in this case. How many people do you have in stock investor circle that are not in the overlapped area? Simply 650-400 or 250 people. Likewise, how many people do we have in the bond investor circle that are not in the overlapped area? Again it is the total number of bond investors, 550, less the 400 in the common area or 550-400 = 150 people.
Now let's look at your Venn diagram. You have an oval containing 1000 investors (the maximum of any calculation). You have two overlapping circles. One represents the stock investor, 250 of which only buy stock and 400 of whom buy stocks and bonds. The other circle represents the bond investors, 150 of whom only buy bonds and 400 of whom buy both.
We can now answer the questions:
a) How many invested in only stocks? The answer is 650-400 = 250
b) How many invested in stocks or bonds? The answer is 250 plus 150 = 400
c) How many did not invest in stocks or bonds? Refer back to the first paragraph above, without overlap we have 650+550 = 1200 investors, more than our total population of people interviewed. We must subtract the overlap (techniquely called the intersection of the two sets) of 400 from this 1200 to yield 800 total investors. Therefore we get 1000 less 800 or 200 non-investors.
We can also solve this problem analytically.
Let S be the set of stock investors, B the set of bond investors and U the universe of all members (investors and nom-investors)
Set of investors = S + B - S(intersect)B
Set of investors = 650 + 550 - 400 = 800
Set of stock DISABLED_event_only= S -S(intersect)B = 650 - 400 = 250
Set of bond DISABLED_event_only= B - S(intersect)B = 550 - 400 = 150
Set of non-investos = U - Set of investors = 1000 - 800 = 200
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