# 9.8.2. Complementary Events and the Difference between Two Events

A card is drawn randomly from 85 identical cards numbered from 1 to 85, find the probability that the number on the drawn card is NOT a perfect square.

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### Example

A card is drawn randomly from 85 identical cards numbered from 1 to 85, find the probability that the number on the drawn card is NOT a perfect square.

### Solution

The probability of event occurring is equal to the number of favorable outcomes divided by the total number of possible outcomes in the sample space, . In this case, event is drawing a card that is a perfect square, so event is drawing a card that is not a perfect square. The cards that are perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, and 81, so these are the favorable outcomes for event . Find the probability of event as follows:

The events and are complementary, so the sum of their probabilities is equal to 1. Now use this fact to find the probability of .

Thus, the probability of drawing a card that is not a perfect square is .

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