Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

One card is selected at random from a standard deck of 52 playing cards. Use a formula to find the probability of the union of the two events. The card is a 5 or a 2

Step-by-Step Solution

Answer

The probability that the card drawn is either a 5 or a 2 is \( \frac{2}{13}\)

Step 1: Calculate Individual Probabilities

First, calculate the probability of drawing a 5 and the probability of drawing a 2 individually. Since there are 4 '5's and 4 '2's in a deck of 52 cards, the probability of drawing a 5 is \(\frac{4}{52}\) or \(\frac{1}{13}\), and the probability of drawing a 2 is also \(\frac{4}{52}\) or \(\frac{1}{13}\).

Step 2: Applying the Union Rule

The probability of the union of two events is given by the formula: \(P(A ∪ B) = P(A) + P(B) - P(A ∩ B)\). Since events A (drawing a 5) and B (drawing a 2) are mutually exclusive (there is no intersection), the formula simplifies to \(P(A ∪ B) = P(A) + P(B)\). Substitute \(\frac{1}{13}\) for both P(A) and P(B).

Step 3: Calculate the Result

Add the two probabilities together: \(\frac{1}{13} + \frac{1}{13} = \frac{2}{13}\). This is the probability that the card drawn is either a 5 or a 2.

Key Concepts

Understanding Mutually Exclusive Events

When dealing with the probability of events, it's crucial to understand the concept of mutually exclusive events. These are events that cannot occur at the same time. In other words, if one event happens, the other cannot. For example, when flipping a coin, the events 'heads' and 'tails' are mutually exclusive because the coin cannot land on both sides simultaneously.

In the context of our playing card problem, the events 'drawing a 5' and 'drawing a 2' are also mutually exclusive. There is no single card in a standard deck that can be both a 5 and a 2 at the same time. Recognizing mutually exclusive events is essential because it simplifies the calculation of the probability of their union.

Applying Probability Formulas

Finding the probability of the union of two events often involves using specific probability formulas. One of the most important formulas is the union rule, described as:

\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).

The term \(P(A \cup B)\) represents the probability of either event A or event B occurring. If events A and B are mutually exclusive, as seen with our 'drawing a 5' and 'drawing a 2' from a deck of cards, the formula simplifies to \(P(A \cup B) = P(A) + P(B)\) since the probability of the intersection of A and B, denoted as \(P(A \cap B)\), is zero. In terms of ease of learning, remembering this rule and identifying when events are mutually exclusive can significantly simplify calculations.

Grasping Basic Probability

Basic probability is the likelihood or chance of an event occurring and is a fundamental concept in statistics. Probability values range from 0 (the event will not occur) to 1 (the event is certain to occur). Probabilities can also be expressed as fractions or percentages.

To calculate the basic probability of a single event, you divide the number of favorable outcomes by the total number of possible outcomes. In our card example, each type of card (2 or 5) has four favorable outcomes (one for each suit), and there are 52 possible outcomes in total. Therefore, the probability of drawing a 2 or a 5 is \(\frac{4}{52}\), which simplifies to \(\frac{1}{13}\). Basic probability principles form the bedrock of more complex probabilistic reasoning and are essential for understanding real-world scenarios involving chance and uncertainty.