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When acting on loan applications it can be concluded, based on historical records, that loan applicants having certain combinations of features can be expected to repay their loans and those who have other combinations of features cannot. As their main features, suppose that a bank uses:

Marital Status: Married, Single (never married), Single (previously married).

Past Loan: Previous default, No Previous default

Employment: Employed, Unemployed (within 1 year), Unemployed (more than 1 year).

(a) How many different loan applications are possible when considering these features?

(b) How many manifestations of loan repayment/default are possible when considering these features?

For part a, it seems plain to me that it's the product rule and is 3 * 2 * 3 = 18. But I'm confused about the wording in (b) - what are manifestations , and how do I approach this part? I appreciate any tips or advice.

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    I think they mean that there are $2^{18}$ ways of drawing up the truth table of marital status / past loan / employment vs whether the load will be repaid or defaulted.2012-02-29
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    oops You are right, Thank You Very Much. Hmm, I'll think more!2012-02-29
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    @David: Just $2\cdot 18=36$, two for each of the $18$ possible application types.2012-02-29
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    @Adel: If @David’s interpretation is right, you’re just adding a fourth category, Repay/Default, with two possible outcomes, so you can use the same kind of reasoning that you used to get $3\cdot 2\cdot 3$ in (a).2012-02-29
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    @BrianM.Scott - either you've interpreted "manifestations" differently from me, or I haven't explained myself clearly. The question said that for some of the 18 possible combinations of features, the loan could be expected to be repaid; and for others, the loan could be expected to be defaulted. There are $2^{18}$ ways to choose which combinations of features resulted in an expectation of repayment. I guess you could argue that the wording of the question implies that at least one combination results in repayment, and at least one combination results in default, making the answer $2^{18}-2$.2012-02-29
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    @David: There are not $2^{18}$ ways to choose a combination of features: there are only $18$. You can have only one feature from each of the three types, Marital Status, Past Loan, and Employment. In other words, I’m basically accepting your interpretation of *manifestations* but arguing that you’re counting incorrectly.2012-02-29
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    @Brian, No I can count. You and I are just understanding "manifestation" differently. I took it to mean the whole table that the bank has, that lists the 18 combinations, and which of them leads to repayment, and which leads to default. There are $2^{18}$ possible tables. But the more I think about it, the more I think you're right about the meaning - here, "manifestation" probably just means one line of the table, that is, just one combination of marital status / loan history / employment. I'm not disputing that there are 18 of these.2012-02-29
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    @David: Sorry $-$ I finally figured out what you’re doing. I should have realized earlier ($2^{18}$ is not $2^8). That possibility just never occurred to me.2012-03-01
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    @Brian - no worries. I just hope that I haven't confused the original poster too much with my misinterpretation and our subsequent discussion.2012-03-01

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It seems to me that the reasonable interpretation is that for every one of the $18$ classes that a particular person can fall into, there are $2$ possibilities for the actual outcome (default or not default), for a total of $36$.

What I would add is that the usage of the word "manifestation" in the question has a definite French tinge. Although using "manifestation" (pronounce it in English) here is not quite idiomatic, if you pronounce it in French, it is perfectly idiomatic.

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    Thank You Very Much! This makes a lot of sense now!2012-03-03