Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Discrete mathematics, continuous mathematics and probability / 8 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

A partition of a set consists of nonempty subsets that are pairwise disjoint and whose union is the whole set. Here

B_{i} \neq = \emptyset, B_{i} \cap B_{j} = \emptyset (i \neq = j), i \in I ⋃ B_{i} = Ω.

Thus every outcome belongs to exactly one cell. The probability calculation additionally requires the cells to be measurable events.

ii

Solution

For a finite or countable measurable partition, the law of total probability is

P (A) = i : P (B_{i}) > 0 \sum P (A ∣ B_{i}) P (B_{i}) .

Indeed,

A

is the disjoint union of

A \cap B_{i}

, and the conditional probability formula gives each positive-probability term. Zero-probability cells contribute zero; their conditionals need not be defined by a ratio. If all cells have positive probability, the sum simply runs over every

i \in I

.

 Ancestors (11)

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version