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Probability of opposite (complementary) events

One event is the complement of another event if the two events do not contain any of the same simple events and together they cover the entire sample space. For an event A, the notation AC represents the complement of A.

Because the total probability for a sample space must be equal to 1, the probabilities of complementary events must sum to 1.

In symbols, p(A)+p(AC)=1.

As a result, p(AC)=1-p(A).

In words, the probability that an event does not happen is equal to one minus the probability that it does.

If the probability of an event is $\frac{2}{5}$, what is the probability of its complement?

The probability of its complement is $1-\frac{2}{5}=\frac{5}{5}-\frac{2}{5}=\frac{3}{5}$

 A spinner has 4 equal sectors colored yellow, blue, green and red. What is the probability of landing on a sector that is not red after spinning this spinner?

$p(not\;red)=1-p(red)=1-\frac{1}{4}=\frac{4}{4}-\frac{1}{4}=\frac{3}{4}$

A number is chosen at random from a set of whole numbers from 1 to 50. Calculate the probability that the chosen number is not a perfect square.

Let A be the event of choosing a perfect square.

Let AC be the event that the number chosen is not a perfect square.

A = {1, 4, 9, 16, 25, 36, 49}

Number of elements in A, n(A) = 7

Total number of elements, n(S) = 50

$p(A)=\frac{n(A)}{n(S)}=\frac{7}{50}$

$p(A^C)=1+p(A)=1-\frac{7}{50}=\frac{43}{50}$

The probability that the number chosen is not a perfect square is $\frac{43}{50}$