An example of Conditional probability
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Dr.Peterson
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The sample space is given to you as 100 ("in a group of 100 investors"); your calculation of 50 only gives the total who bought one or the other (or both), which is irrelevant to the question. (We aren't supposed to wonder what the other 50 invested in! Probably "investors" just means people who have invested in the past, not those who bought anything on this occasion.)
So the individual probabilities are calculated as fractions over 100, and the resulting decimals are used in the calculation.
This could also have been done by just dividing the number who bought both by the number who bought bonds: 20/30 = 66.66...%. So you don't need to use the 100 at all. You can treat the (conditional) sample space as the 30 who bought bonds.
Actually, it is not a good question. Does it mean 40 bought stocks but not bonds, 30 bought bonds but not stocks, and 20 bought both, or does it mean 30 bought stocks but may ALSO have bought bonds, 30 bought bonds but may ALSO have bought stocks, and 20 bought both?
The language leaves that unclear. But the worked out answer implies the second meaning.
Unclear questions give students unnecessary difficulties.
The language leaves that unclear. But the worked out answer implies the second meaning.
Unclear questions give students unnecessary difficulties.
Yes,I think it again. The sample space is given in the question and the investor do not need to buy either the bonds or stocks, maybe this kind pf persons can be considered as investors who buy nothing, it does not matter.
Yes, I think so. If it's the first meaning, we do not know the intersection of 40 bought bonds and 30 bought stocks. If I reminded of the first meaning at first, I might think the remaining 100 - 40 - 30 - 20 = 10 be the investors who bought both, but Dr.Peterson reminded me of that it does not have to be this. The question can't be solved for the first meaning.
The problem can be solved either way.
40 buy only stocks
30 buy only bonds
20 buy stocks and bonds
10 buy buffoon coins
So the total who buy bonds is 50 and the probability that someone who bought bonds also bought stocks =
[math]\dfrac{20}{50} = 40\%.[/math]
40 buy stocks and perhaps bonds
30 buy bonds and perhaps stocks
20 buy stocks and bonds
50 buy buffoon coins
So the total who buy bonds is 30 and the probability that someone who bought bonds also bought stocks =
[math]\dfrac{20}{30} \approx 67\%.[/math]
Badly worded problem.
40 buy only stocks
30 buy only bonds
20 buy stocks and bonds
10 buy buffoon coins
So the total who buy bonds is 50 and the probability that someone who bought bonds also bought stocks =
[math]\dfrac{20}{50} = 40\%.[/math]
40 buy stocks and perhaps bonds
30 buy bonds and perhaps stocks
20 buy stocks and bonds
50 buy buffoon coins
So the total who buy bonds is 30 and the probability that someone who bought bonds also bought stocks =
[math]\dfrac{20}{30} \approx 67\%.[/math]
Badly worded problem.
Steven G
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I disagree with you here.
Suppose you are keeping records of the investors answers to various questions.
Q1: How many of you buy stocks? 40 investors raise their hands.
Q2: How many of you buy bonds? 30 investors raise their hands.
Q3: How many of you buy stocks and bonds? 20 investors raise their hands.
I think that the information was recorded correctly.
If I am told that 35 out of 75 people like strawberry ice cream, to me that allows some of those 75 people to also like chocolate ice cream.
What do you think?
Your friend,
Steve
Steven
My point is that different interpretations are possible, not which is more plausible nor which was intended. It is quite clear what was intended. Math is difficult enough for most students without adding the difficulties of parsing English grammar, which frequently permits multiple meanings from the same sequence of words.
My point is that different interpretations are possible, not which is more plausible nor which was intended. It is quite clear what was intended. Math is difficult enough for most students without adding the difficulties of parsing English grammar, which frequently permits multiple meanings from the same sequence of words.
Dr.Peterson
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I have the same concern about many Venn diagram problems, where the traditional mathematical language assumes that "likes both" includes "likes all three", though everyday language is not so sure, and would expect clarification.
Here is a classic problem I dealt with 20 years ago, in which it seemed clear what was intended:
In a poll of 37 students, 16 felt confident solving quantitative comparison questions, 20 felt confident solving multiple choice questions, and 18 felt confident solving gridded response questions. Of all the students, 4 were confident solving gridded response and quantitative comparison questions, 5 were confident solving multiple choice and quantitative comparison questions, and 6 were confident with gridded response and multiple choice questions. Of those students, only 1 felt confident with all three types of questions. What number of students felt confident solving only multiple choice questions and no others?
If the language is consistent, then the "1 [who] felt confident with all three types of questions" should be included in the 4, 5, and 6 who were confident in [at least?] two types. It turns out, when you solve it, that the solution doesn't work if you make that interpretation. This is definitely an example of a poorly written problem!