Complement, Independence, Mutually Exclusive
Complement and Mutually Exclusive
Consider a course where, if completed, you can receive one of the following letter grades:
A, B, C, D, or F and each has an equal probability of occurring. That is, P(A) – read
probability of event A – is 0.20; P(B) = 0.20; and so on for the remaining three outcomes.
Question: If we define event A as {getting an A in the course}, what is the complement
of A (written Ac)?
Answer: The complement of event A would mean all outcomes not in event A. Thus the
events in Ac would be {getting a B, or a C, or a D, or an F}. So don’t think that just
getting a B would be the complement of A; the complement is all outcomes not in event
A.
Question: Are the two events A = {getting an A in the course} and B = {getting a B in
the course} mutually exclusive?
Answer: Yes, these two events are mutually exclusive since they cannot occur at the
same time: you cannot receive both an A and a B for the same course. Think of this
current course; at the end of the semester you cannot be given an A and a B.
Question: Are the two events A and B independent?
Answer: There are three formulas, from which we only need to use one since the three
are intertwined, meaning that if any one shows independence (or dependence) then the
other two will follow. We can show independence exists by satisfying any one of:
1. If two events, say event A and event B, are independent then P(A)*P(B) = P(A
and B). In this example, P(A)*P(B) = (0.20)*(0.20) = 0.04, but the P(A and B) –
that is the probability both events A and B occur together – is equal to 0, i..e. P(A
and B) = 0. Since P(A)*P(B) = 0.04 does not equal P(A and B) = 0 then events A
and B are not independent, thus events A and B are dependent. This is true of any
events that are mutually exclusive.
2. If two events, say events A and B, are independent then P(A|B) = P(A). By
P(AandB )
formula, P(A|B) =
and as shown above in 1, P(A and B) = 0 meaning
P(B)
P(A|B) = 0. But we know that P(A) = 0.20. Therefore, P(A|B) does not equal
P(A) so again events A and B are not independent.
3. If two events, say events A and B, are independent then P(B|A) = P(B). By
P(AandB )
formula, P(B|A) =
and as shown above in 1, P(A and B) = 0 meaning
P( A)
P(B|A) = 0. But we know that P(B) = 0.20. Therefore, P(B|A) does not equal
P(B) so again events A and B are not independent.
As you can see, all three mathematical methods reach the same conclusion. That is why
you only need to select one of these formulas in order to determine whether or not the
events are independent. NOTE: P(A|B) is read “Probability that event A occurs given
that event B occurs.”
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Independence
The concept of independence means that knowing the outcome of one event does not
change the probability that another event occurs.
For instance, consider the probability that a randomly selected teenager enjoys shopping
is 0.5. That is, if we let event S = {teenager enjoys shopping}, then P(S) = 0.5. But what
if we know that teenage girls enjoy shopping more so than teenage boys (this is
hypothetical I am not stereotyping!). Then if we knew the gender of the randomly
selected teenager the probability that they enjoy shopping would increase if that selected
teenager were a girl and decrease if that teenager were a boy. Therefore, if we let event
G = (gender of teenage is a girl) the two events S and G would be dependent, i.e. not
independent because knowing (i.e. “given”) that the randomly selected teenager is a girl
increases the probability that this teenager enjoys shopping. CAUTION: this does not
mean to imply that the teenager does enjoy shopping, but only that the probability that
the teenager enjoys shopping increases knowing that the teenager is a girl. The teenager
either does or does not enjoy shopping. Think of probability as the before the truth
likelihood that an event occurs. For example, say the probability that the Penn State
football team wins its first game of the season is 0.85, but after that first game is played,
Penn State has either won or lost the game. So the probability is just the likelihood that
an event occurs and does not mean that event will happen (unless of course the
probability of the event is 1).
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