Question

if j, i and l and not mutually exclusive events, what is p(j or k or l)? Explain how this formula is found .

Answer 1

p(j or k or l) = p(j) + p(k) + p(l) - p(j ∩ k) - p(k ∩ l) - p(l ∩ j) + p(j ∩ k ∩ l)

Explanation:

First let j ∪ k = m.

Then, p(j or k or l) = p(j ∪ k ∪ l) = p(m ∪ l)

Here, m and l are not mutually exclusive.

However, we can write the m ∪ l as a union of two disjoint sets as below:

m ∪ l = m ∪ (m' ∩ l)

where m and m' ∩ l are mutually exclusive.

Now, p(m ∪ l) = p[m ∪ (m' ∩ l)]

= p(m) + p(m' ∩ l)

p(m' ∩ l) = p(l) - p(m ∩ l)

So, p(m ∪ l) = p(m) + p(l) - p(m ∩ l) --- (1)

Now, p(m) = p(j ∪ k)

= p(j) + p(k) - p(j ∩ k) from the result of (1)

Also, p(m ∩ l) = p[(j ∪ k) ∩ l]

= p(j ∩ l) ∪ p(k ∩ l)

= p(j ∩ l) + p(k ∩ l) - p[(j ∩ l) ∩ (k ∩ l)]

= p(j ∩ l) + p(k ∩ l) - p(j ∩ k ∩ l)

Hence, (1) becomes

p(j ∪ k ∪ l) = p(j) + p(k) + p(l) - p(j ∩ k) - p(k ∩ l) - p(l ∩ j) + p(j ∩ k ∩ l)