Two events: “Mutually Exclusive” or “Independent” or Both or Neither?
Assume that we are working with two non-empty events, 𝐴 and 𝐵. That is, 𝑃(𝐴) > 0 and 𝑃(𝐵) > 0.
Recall the definitions and formulas
Mutually Exclusive
Mutually Exclusive means that 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 0.
Recall the Multiplication Formula: 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵|𝐴) = 𝑃(𝐵) ∙ 𝑃(𝐴|𝐵)
Independent
Independent means that 𝑃(𝐵|𝐴) = 𝑃(𝐵|𝑛𝑜𝑡 𝐴) = 𝑃(𝐵) and 𝑃(𝐴|𝐵) = 𝑃(𝐴|𝑛𝑜𝑡 𝐵) = 𝑃(𝐴).
Cannot be both
If two events are Mutually Exclusive, they cannot be Independent, too
Proof: Assume that events 𝐴 and 𝐵 are mutually exclusive and that each has nonzero probability. That
is, 𝑃(𝐴) > 0 and 𝑃(𝐵) > 0.
Since events 𝐴 and 𝐵 are mutually exclusive,
The Multiplication Formula lets us replace
𝑃(𝐴 𝑎𝑛𝑑 𝐵)
Since 𝑃(𝐴) > 0, we can divide each side of the
equation by 𝑃(𝐴). Likewise for 𝑃(𝐵) > 0.
Simplify each side of the equation
If they were independent events, we would have:
Transitive Property of Equality then says:
Contradiction!
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 0
𝑃(𝐴) ∙ 𝑃(𝐵|𝐴) = 0
𝑃(𝐵) ∙ 𝑃(𝐴|𝐵) = 0
𝑃(𝐴) ∙ 𝑃(𝐵|𝐴)
0
=
𝑃(𝐴)
𝑃(𝐴)
𝑃(𝐵|𝐴) = 0
𝑃(𝐵|𝐴) = 𝑃(𝐵)
𝑃(𝐵) = 0
Contradicts 𝑃(𝐵) > 0
𝑃(𝐵) ∙ 𝑃(𝐴|𝐵)
0
=
𝑃(𝐵)
𝑃(𝐵)
𝑃(𝐴|𝐵) = 0
𝑃(𝐴|𝐵) = 𝑃(𝐴)
𝑃(𝐴) = 0
Contradicts 𝑃(𝐴) > 0
1. Since they are Mutually Exclusive, 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 0
2. Multiplication Formula:
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 0 = 𝑃(𝐴) ∙ 𝑃(𝐵|𝐴) and also 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐵) ∙ 𝑃(𝐴|𝐵)
3. Since 𝑃(𝐴) > 0 and 𝑃(𝐵) > 0, we can divide each side of the first equation by 𝑃(𝐴) and we can
divide each side of the second equation by 𝑃(𝐵). This gives 0 = 𝑃(𝐵|𝐴) and 0 = 𝑃(𝐴|𝐵).
4. But if the events were independent, we have 0 = 𝑃(𝐵|𝐴) = 𝑃(𝐵) and 0 = 𝑃(𝐴|𝐵) = 𝑃(𝐴).
Contradiction! We had assumed that 𝑃(𝐴) > 0 and that 𝑃(𝐵) > 0.
If two events are Independent, they cannot be Mutually Exclusive, too
Proof: Assume that events 𝐴 and 𝐵 are independent and that each has nonzero probability. That is,
𝑃(𝐴) > 0 and 𝑃(𝐵) > 0.
Since events 𝐴 and 𝐵 are independent,
Multiply each side of the first equation
by 𝑃(𝐴). Multiply each side of the
second equation by 𝑃(𝐵).
The left side of each of these is the
same as 𝑃(𝐴 𝑎𝑛𝑑 𝐵), by the
Multiplication Formula
Since 𝑃(𝐴) > 0 and 𝑃(𝐵) > 0, we
must have 𝑃(𝐴 𝑎𝑛𝑑 𝐵) > 0.
Conclusion: Not Mutually Exclusive
𝑃(𝐵|𝐴) = 𝑃(𝐵)
𝑃(𝐴) ∙ 𝑃(𝐵|𝐴) = 𝑃(𝐴) ∙ 𝑃(𝐵)
𝑃(𝐴|𝐵) = 𝑃(𝐴)
𝑃(𝐵) ∙ 𝑃(𝐴|𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵)
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵)
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵) > 0
𝑃(𝐴 𝑎𝑛𝑑 𝐵) > 0
Because if they were mutually exclusive, 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 0.
Two Events can be Mutually Exclusive and Not Independent
Proof by example – one trial, draw a card, events 𝐾 = king, 𝑄 = queen
Mutually Exclusive: 𝑃(𝐾 𝑎𝑛𝑑 𝑄) = 0
1
But Not Independent: 𝑃(𝐾) = 13 ; 𝑃(𝐾) ≠ 𝑃(𝐾|𝑄) = 0
and for emphasis, 𝑃(𝑄) =
1
; 𝑃(𝑄)
13
≠ 𝑃(𝑄|𝐾) = 0
Two Events can be Independent but not Mutually Exclusive
Example – one trial, draw a card, events 𝐾 = king, 𝑅 = red.
1
Not mutually exclusive: 𝑃(𝐾 𝑎𝑛𝑑 𝑅) = 52
Independent: 𝑃(𝐾) =
1
13
1
2
= 𝑃(𝐾|𝑅) and 𝑃(𝑅) = = 𝑃(𝑅|𝐾)
Two Events can be neither Independent nor Mutually Exclusive
Example – one trial, draw a card, events 𝐻=higher than a 5 (six thru ace), 𝐿 =lower than 10 (two thru
9
7
nine). 𝑃(𝐻) = 13; 𝑃(𝐿) = 13.
4
Not mutually exclusive: 𝑃(𝐻 𝑎𝑛𝑑 𝐿) = 13 (six, seven, eight, nine)
4
1
Not independent: 𝑃(𝐻|𝐿) = 8 = 2 (𝐿 narrows it down to eight possible ranks, four of those fit the 𝐻)
4
And similarly for 𝑃(𝐿|𝐻) = 9 ≠ 𝑃(𝐿).