# 11.4.1. Calculating Probability

A card is selected randomly from 52 cards numbered from 1 to 52, if the number written on it is recorded. Find the probability that the number written on the selected card is not a perfect square number and not a multiple of number 6.

• A
• B
• C
• D

### Example

If a card is drawn randomly from a deck of 52 cards numbered from 1 to 52, determine the probability that the number on the drawn card is not a perfect square and not a multiple of 6.

### 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. The cards that are perfect squares are 1, 4, 9, 16, 25, 36, and 49, so these are the favorable outcomes for event . Find the probability of event as follows:

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 multiple of 6. The cards that are multiples of 6 are 6, 12, 18, 24, 30, 36, 42, and 48, so these are the favorable outcomes for event . Now find the probability of event .

The notation refers to the probability of either event or event occurring, or both. If and are two non-mutually exclusive events, then , where the notation refers to the probability of events and occurring together. The only card that is both a perfect square and a multiple of 6 is 36, so is equal to . Next, use this fact to help find .

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

De Morgan's First Law is , where the notation means the non-occurrence of and the non-occurrence of . Since , it follows that . Thus, the probability that the card drawn is not a perfect square and not a multiple of 6 is .

0
correct
0
incorrect
0
skipped