cepheid said:
So which answer is correct? I'm curious.
What if the question asked you to "write down a number between one and ten" and you wrote
four but the answer “in the book” was
six, who's correct? It's quite simply an incompletely specified problem.
In this case
P(A) and
P(B) alone just don't fully specify
P(A and B) without further assumptions.
In particular,
P(A or B) = P(A) + P(B) - P(A and B)
and also,
P(A and B) = P(A | B) P(B) = P(B | A) P(A)
where
P(A | B) reads “Probability of event
A occurring given that event
B has occurred”.
So just knowing
P(A) and
P(B) is not enough, you must also know either
P(A or B) or one of the two conditional probabilities
P(A | B) or
P(B | A) in order to fully specify it.
The conditional probably
P(A | B), like any probability, can take values between zero and one. The extreme case of
P(A | B) = 0 corresponds to “mutually exclusive” events where in this case the occurrence of event
B totally precludes event
A from occurring. The other extreme case of
P(A | B) = 1 corresponds to
A and
B being so closely related that
A must always occur when
B occurs (ie,
B a subset of
A).
In general when
P(A | B) has a value that is greater then
P(A) then it means that there is some type of positive correlation between event
B and event
A. Conversely when
P(A | B) is less than
P(A) then there is some sort of negative correlation, where event
B occurring inhibits the chances of event
A happening. The other salient position is an intermediate case where
P(A | B) is equal to
P(A) and thus the occurrence or otherwise of event
B has no impact at all on the chances of event
A. This is the case of “independent events”.
So in this question there are a whole range of potential answer for
P(A and B) depending on what assumption you make about the inter-dependence of the English results and the Maths results.
The extreme cases (that are consistent with the given data) can be found as follows.
1. Clearly
P(A and B) <= Min( P(A), P(B) ), so
P(A and B) is less than or equal to
10/15.
2.
P(A and B) = P(A) + P(B) - P(A or B), whih implies that
P(A and B) >= P(A) + P(B) -1.
So in this case
P(A and B) >= 10/15 + 12/15 - 1 =
7/15.
Case 1 above (10 students pass both subjects) corresponds to the most optimistic assumption that we can make (while remaining consistent with the given data) regarding the correlation between passing Maths and passing English. Case 2 on the other hand (7 students pass both subjects) corresponds to the most pessimistic (data compatible) assumption and in this scenario we're actually implying that passing English inhibits your chances of passing Maths and visa versa.
The intermediate assumption of independence corresponds to
P(A and B) = P(A) P(B) giving a resultant probability of
8/15 (8 students pass both subject).
Possible answers range from
7 through to
10 inclusive, and any answer to this question should have an assumption stated. I'd give zero marks for an answer of 7 without any supporting assumptions.
