# Find the probability that the first card dealt off the top of a deck of cards is a diamond or a...

## Question:

Find the probability that the first card dealt off the top of a deck of cards is a diamond or a 5. (There are thirteen diamonds and four 5's in a 52-card deck of cards, including one 5 of diamonds.)

## Probability of Non-Mutually Exclusive Events

When we want to find the probability that either one event or another occurs, we can only say that this is the sum the individual probabilities if these events are mutually exclusive. If there are any cases in which both events could simultaneously occur, we need to subtract these duplicates from this sum.

In order to determine this probability, we first need to determine whether these events are mutually exclusive. Since there is a five of diamonds, there one card that is both a diamond and a five. Thus, these events are not mutually exclusive, and so we need to be sure to subtract off duplicates in finding this probability. The formula that we use to find this probability is:

{eq}\displaystyle P(5 \text{ or diamond }) = P(5) + P(\text{ diamond }) - P( 5 \text{ and diamond }) {/eq}

Out of the 52 card deck, four of the cards are fives, thirteen are diamonds, and one card is the five of diamond. Thus, we can find this probability as follows.

{eq}\displaystyle P(5 \text{ or diamond }) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} {/eq} 