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Past exam of the mathematics course of the University of Cambridge
/
2021
/
ia
/
Paper 2
/
9E
/
a
/
i
/
Solution
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2021
ia
Paper 2
9E
a
i
Created
2026-09-24
Updated
2026-09-25
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For
P
(
B
)
>
0
, the
conditional probability
is
P
(
A
∣
B
)
=
P
(
B
)
P
(
A
∩
B
)
.
(1)
Hence
P
(
B
j
∣
A
)
=
P
(
A
)
P
(
A
∣
B
j
)
P
(
B
j
)
.
(2)
Because the
B
k
form
a
partition, the
law
of total
probability
gives
P
(
A
)
=
∑
k
=
1
n
P
(
A
∣
B
k
)
P
(
B
k
)
.
(3)
Substitution proves
Bayes theorem
in the required form:
P
(
B
j
∣
A
)
=
∑
k
=
1
n
P
(
A
∣
B
k
)
P
(
B
k
)
P
(
A
∣
B
j
)
P
(
B
j
)
.
(4)
Ancestors
(12)
i
a
9E
Paper 2
ia
2021
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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