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

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)

Step-by-step 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)