Problem 43 - College Algebra with Corequisite Support

Question

For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following: A head on the coin or a club

Step-by-Step Solution

Answer

The probability of getting a head on the coin or drawing a club is \(\frac{5}{8}\).

Step 1: Understand the Event Categories

The problem involves two independent actions: 1) tossing a coin and 2) drawing a card from a standard deck. These events are independent because the outcome of the coin toss does not affect the card drawn.

Step 2: Identify Possible Outcomes

A coin toss has two outcomes: 'heads' (H) or 'tails' (T). A standard deck has 52 cards, including 13 clubs.

Step 3: Calculate Individual Probabilities

The probability of getting a head is \(P(H) = \frac{1}{2}\). The probability of drawing a club is \(P(C) = \frac{13}{52} = \frac{1}{4}\).

Step 4: Identify Overlapping Events

Evaluate if there's any event that results in both a 'head' and a 'club'. Since these are independent, we don't need to account for overlaps as described, contrary to dependent events.

Step 5: Apply the Rule for 'Or' Probability

Since there's no overlap between getting a head and getting a club, use the formula: \[ P(H \text{ or } C) = P(H) + P(C) - P(H \cap C) \] where \(P(H \cap C) = P(H) \cdot P(C) = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}\).

Step 6: Calculate the Final Probability

Substitute the known probabilities into the formula and solve:\[ P(H \text{ or } C) = \frac{1}{2} + \frac{1}{4} - \frac{1}{8} \]Convert the terms to a common denominator:\[ P(H \text{ or } C) = \frac{4}{8} + \frac{2}{8} - \frac{1}{8} = \frac{5}{8} \].

Key Concepts

Independent Events

In probability theory, independent events are those whose occurrence or outcomes do not affect each other. Consider flipping a coin and selecting a card from a deck. No matter the result of the coin flip – whether heads or tails – it doesn’t change the nature or probability of drawing a club from the deck. Because of this independence, the calculations for probabilities can be treated separately for each event. This characteristic is crucial when determining the likelihood of combined events, as independent events allow for the clear application of probability rules, without needing to consider how one event might influence another.

Probability Rules

Probability rules help us calculate the chances of one or more events happening. When dealing with independent events, these rules become straightforward. For example, when we want to find the probability of either a coin landing heads or drawing a club, we apply the rule of addition for probabilities. This rule states that the probability of one event or another happening is the sum of their individual probabilities, minus the probability that both happen together (the overlap). Because our events are independent, calculating the overlap is as simple as multiplying the individual probabilities together.

• Addition Rule: Adjusts for overlap by subtracting the joint probability.
• Multiplication Rule: Used for overlapping probability of two independent events.

Understanding these rules allows you to accurately compute complex probability questions.

Sample Space

Sample space is a fundamental concept in probability, referring to all the possible outcomes of a particular experiment. For a coin toss, the sample space can be simply described as \("HEAD" or "TAIL"\) – two possible outcomes. For a standard deck of cards, however, the sample space expands. It contains 52 potential outcomes, 13 of which are clubs. Having a comprehensive understanding of the sample space allows you to correctly assign probabilities to different events. This understanding helps avoid overlooking any possible outcomes when calculating probabilities for each scenario. And it ensures that probabilities add up to 1, as all potential events are accounted for in the sample space.

Complementary Events

Complementary events are pairs of outcomes that together encompass all possibilities for a given scenario. They are mutually exclusive and collectively exhaustive. For example, getting heads is the complement of getting tails in a coin toss – together, they make up the entire sample space for that single event. An understanding of complementary events helps in probability calculation, because the probability of an event not occurring is simply one minus the probability of it occurring.
When dealing with coin flips and card draws, identifying and calculating the probability of complementary events can sometimes simplify the problem and offer useful shortcuts. This is because once you know the probability of one event, you can instantly infer the probability of its complement.