Problem 12 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

An experiment consists of selecting a card at random from a 52-card deck. Refer to this experiment and find the probability of the event. A black face card is not drawn.

Step-by-Step Solution

Answer

The probability of not drawing a black face card from a standard 52-card deck is \(\frac{23}{26}\) or approximately 0.885.

Step 1: Determine the number of black face cards in the deck

A standard 52-card deck consists of 4 suits: hearts (red), diamonds (red), clubs (black), and spades (black). There are 3 face cards in each suit: Jack, Queen, and King. Since there are 2 black suits (clubs and spades), there are 2 × 3 = 6 black face cards in the deck.

Step 2: Find the total possible outcomes

A 52-card deck has 52 cards, so there are 52 possible outcomes when drawing one card.

Step 3: Find the probability of drawing a black face card

The probability of an event is given by the number of favorable outcomes divided by the total possible outcomes. In this case, the number of favorable outcomes is 6 (black face cards) and the total possible outcomes is 52. So, the probability of drawing a black face card is: \(P(\text{black face card}) = \frac{6}{52}\)

Step 4: Find the complement of the probability

The complement of an event is the probability of the event not happening. To find the complement of the probability of drawing a black face card, we can subtract the probability of drawing a black face card from 1, since the probability of all possible outcomes is always 1. Therefore, the probability of not drawing a black face card is: \(P(\text{not black face card}) = 1 - P(\text{black face card})\) \(P(\text{not black face card}) = 1 - \frac{6}{52}\)

Step 5: Simplify the probability

Now, we simplify the expression for the probability of not drawing a black face card: \(P(\text{not black face card}) = \frac{52}{52} - \frac{6}{52} = \frac{52-6}{52} = \frac{46}{52} = \frac{23}{26}\) So, the probability of not drawing a black face card is \(\frac{23}{26}\) or approximately 0.885.

Key Concepts

Experimental Probability

Experimental probability is a hands-on approach to probability, where one determines the likelihood of an event by conducting an experiment and recording the actual results. It involves observing how often an event occurs, compared to the total number of trials. For instance, if we want to find the experimental probability of drawing a red card from a deck, we could physically draw a card many times, keep track of the number red cards selected, and then divide by the total number of draws.
This differs from theoretical probability, which is based on the possible outcomes without conducting an experiment. In short, experimental probability is grounded in real-life testing, providing practical insights that complement theoretical predictions.

Complement of an Event

In probability theory, understanding the complement of an event is crucial as it gives us the probability that the event does not occur. The complement is denoted by \(A'\) for an event \(A\), and it's found by subtracting the probability of the event from 1.
In the context of the card experiment, if we wanted to calculate the chance of not drawing a black face card, we consider the complement, since the event and its complement together make up all possible outcomes. We have \(P(\text{not black face card}) = 1 - P(\text{black face card})\). Knowing about complements helps in quickly finding the probability of an event not happening, which often simplifies more complex probability problems.

Mathematical Probability

Mathematical probability is the formal approach to estimating the chance of an event based on the known quantity of possible outcomes. It's represented by the ratio of favorable outcomes to the total possible outcomes. This is done numerically, using the formula \(P(E) = \frac{\text{number of favorable outcomes}}{\text{total possible outcomes}}\), where \(E\) is the event in question.
For example, in a deck of 52 cards, the mathematical probability of drawing a black face card is calculating the number of black face cards (6), and dividing that by the total number of cards (52), resulting in \(P(\text{black face card}) = \frac{6}{52}\). This form of probability is fundamental in statistics and helps students understand the likelihood of various scenarios without the need for physical trials.

Basic Statistics

Basic statistics involves collecting, analyzing, interpreting, and presenting data. It provides tools to understand the data and draw inferences about a population based on a sample. In terms of probability, statistics uses the principles mentioned above to make predictions and decisions. Key statistical concepts include mean, median, mode, range, and standard deviation, which help describe and understand data sets.
Statistics often overlaps with probability, as both are concerned with patterns and inferences. For instance, understanding probability can help predict future events based on past data, an idea central to statistics used in fields ranging from economics to natural sciences. Students learning basic statistics begin with probability, as it lays the groundwork for more advanced we can subtract the probability series of statistical analyses.