Problem 39 - College Algebra

Question

You are dealt one card from a 52 -card deck. Find the probability that you are dealt: a 7 or a red card.

Step-by-Step Solution

Answer

The probability of drawing a 7 or a red card from a 52-card deck is \(\frac{28}{52}\) or approximately 0.54 (54%).

Step 1 - Identify Number of Desired Outcomes

Firstly, identify the number of desired outcomes from the total possible outcomes. There are four 7s (hearts, diamonds, spades, and clubs) and twenty-six red cards (hearts and diamonds). The total number of desired cards can be worked out by adding these together.

Step 2 - Adjust for Overlap

In this case, we can see that two 7s (hearts and diamonds) are also red 7s. This means we have double-counted these cards. Therefore, to correct for this overlap, we must subtract the number of red 7s (2) from our total.

Step 3 - Calculate Probability

Finally, to calculate the probability, we divide the number of desired outcomes, adjusted for overlaps, by the total number of cards (52). The result will represent the probability of drawing a 7 or a red card.

Key Concepts

Combination and Permutation

In probability, understanding the difference between combinations and permutations is crucial. A permutation implies an arrangement or sequence where order matters. For example, arranging three letters A, B, and C into different sequences, like ABC, ACB, etc., shows how order changes the outcome.
In contrast, combinations focus on selection where order does not matter. For instance, choosing two fruit from a group of apple, orange, and banana is the same as choosing apple then banana or banana then apple. Here, the order of selection does not change the group chosen.
In the exercise of drawing cards from a deck, combinations and not permutations are used. That’s because the order in which you draw the cards doesn’t affect the end result. You are interested only in whether the card is a 7 or red, regardless of its position in the deck.

Counting Problems

Counting problems in probability are about determining how many ways an event can occur. The foundation of counting problems is to clarify what you wish to count and then to find methods of counting those things efficiently. This often involves using rules like the addition and multiplication principles.
In the given card problem, the addition principle is primarily used. It helps count the total number of cards that meet either of two conditions: being a 7 or being red.

• First, count the number of 7s: four cards (one for each suit).
• Next, count the number of red cards, made up of hearts and diamonds: twenty-six cards.
• Then notice the overlap (red 7s), which are double-counted: two cards (one red 7 in hearts and one in diamonds).
• Subtract these overlaps from the total desired outcomes.

This method ensures that each card is counted correctly just once.

Card Deck Problems

Probability scenarios often take advantage of familiar examples, such as drawing cards from a standard 52-card deck. It's an ideal practice ground because it clearly defines the sample space and possible outcomes. Understanding card deck composition helps in quickly identifying probabilities.

• First, recognize the deck structure: every suit (clubs, diamonds, hearts, spades) contains 13 cards.
• There are two colors represented: red (hearts, diamonds) and black (spades, clubs).

Solving card deck problems involves calculating probabilities based on these properties. In your problem, you calculate the odds of drawing either a 7 or a red card. It requires counting specific cards (7s and red cards) and dealing with overlap correctly to avoid errors. By knowing the basics of the deck, these solutions become more approachable and straightforward.