Problem 23 - Algebra for College Students

Question

What is the probability of getting either a heart or an ace when drawing a single card from a deck of 52 cards?

Step-by-Step Solution

Answer

The probability is \(\frac{4}{13}\).

Step 1 - Identify the Total Number of Possible Outcomes

There are 52 cards in a standard deck. So, the total number of possible outcomes is 52.

Step 2 - Determine the Number of Hearts

There are 13 hearts in a standard deck of 52 cards.

Step 3 - Determine the Number of Aces

There are 4 aces in the deck, one in each suit.

Step 4 - Identify Overlapping Outcomes

The ace of hearts is both an ace and a heart, so it has been counted twice in the previous steps. There is 1 overlapping card.

Step 5 - Calculate Total Favorable Outcomes

Add the number of hearts and the number of aces, then subtract the overlapping ace of hearts: 13 (hearts) + 4 (aces) - 1 (ace of hearts) = 16 favorable outcomes.

Step 6 - Calculate the Probability

The probability of drawing either a heart or an ace is given by the number of favorable outcomes divided by the total number of possible outcomes: \[ P(\text{Heart or Ace}) = \frac{16}{52} = \frac{4}{13} \]

Key Concepts

Total Outcomes

To understand probability in card games, you first need to know the total number of possible outcomes. In a standard deck of cards, there are 52 cards. These cards are divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, making a total of 52 cards in the deck. When calculating the probability of any event happening, you'll often start with this number of total possible outcomes. This is crucial, as it sets the groundwork for all further calculations.

Favorable Outcomes

Favorable outcomes are the outcomes that fit the conditions of the event you are interested in. If you are drawing a single card and want to find the probability of drawing either a heart or an ace, you need to count all the cards that are either hearts or aces. There are 13 hearts in the deck. Additionally, there are 4 aces. Therefore, you might initially think there are 17 favorable outcomes (13 hearts + 4 aces). But be careful; there could be overlap.

Overlapping Outcomes

Overlap occurs when one outcome fits multiple conditions. In our example, the ace of hearts is both a heart and an ace. We counted it once among the hearts and once among the aces. However, this double-counting leads to an error. There is only one ace of hearts. To avoid this error, you subtract the overlapping outcomes from your total favorable outcomes. So, for our event, we subtract the 1 overlapping card from the 17 initial favorable outcomes, giving us 16 distinct favorable outcomes.

Probability Calculation

Once you've identified the total number of outcomes and the number of favorable outcomes, you can calculate the probability. The probability is the ratio of favorable outcomes to total outcomes. For drawing either a heart or an ace from a deck of 52 cards, there are 16 favorable outcomes and 52 total outcomes. So, the probability is calculated as: \[ P(\text{Heart or Ace}) = \frac{16}{52} = \frac{4}{13} \] This means you have a \frac{4}{13} chance of drawing either a heart or an ace when you draw a single card from a deck.