Eris13696
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I don’t understand exactly what they mean in b) or how they derived the P(only B) and P(only H)
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econuser101
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Hi.

Perhaps it is best to start by decomposing the question at hand. Some students take just history, some take just biology and some take both history and biology, in addition to English (which everyone takes). The question states 'given the student took exactly one of history or biology, find the probability it was history.' As such, we can disregard: P(HnB) - the student only takes one of either history or biology.

The easiest way to visualise this is through a Venn diagram, where one circle represents history, and the other biology. The overlap of these two circles is the intersection, i.e. P(HnB). In this case, P(HnB) = 0.1. The question states 60% took history and 30% took biology. As such, we can find the required probabilities as follows (this is also known as the set difference):

P(H) = 0.6 - P(HnB) = 0.6 - 0.1 = 0.5
P(B) = 0.3 - P(HnB) = 0.3 - 0.1 = 0.2

Having found these probabilities, it is quite simple to find the conditional probability P(H|one subject). This is simply equal to:

P(H) / ((P(H)+P(B)) = 0.5 / (0.5+0.2) = 0.5 / 0.7 = 5/7.

Hopefully this clarifies
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Eris13696
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(Original post by econuser101)
Hi.

Perhaps it is best to start by decomposing the question at hand. Some students take just history, some take just biology and some take both history and biology, in addition to English (which everyone takes). The question states 'given the student took exactly one of history or biology, find the probability it was history.' As such, we can disregard: P(HnB) - the student only takes one of either history or biology.

The easiest way to visualise this is through a Venn diagram, where one circle represents history, and the other biology. The overlap of these two circles is the intersection, i.e. P(HnB). In this case, P(HnB) = 0.1. The question states 60% took history and 30% took biology. As such, we can find the required probabilities as follows (this is also known as the set difference):

P(H) = 0.6 - P(HnB) = 0.6 - 0.1 = 0.5
P(B) = 0.3 - P(HnB) = 0.3 - 0.1 = 0.2

Having found these probabilities, it is quite simple to find the conditional probability P(H|one subject). This is simply equal to:

P(H) / ((P(H)+P(B)) = 0.5 / (0.5+0.2) = 0.5 / 0.7 = 5/7.

Hopefully this clarifies
Thank you so much for taking your time to write this explanation, it’s very helpful!! <3
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