Problem 67 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Assuming the next U.S. president will be a Democrat or a Republican, the probability of a Republican president is 0.5

Step-by-Step Solution

Answer

The statement makes sense. Given there are only two likely options - a Democrat or a Republican, and assuming no other influences, it is reasonable to assume that the probability of a Republican president is 0.5.

Step 1: Understanding the Statement

Given the two outcomes are equally likely, each event should have a probability of 0.5 or 50%. Considering that the political scenario only allows a Democrat or a Republican to become the next U.S. President, the probability for each should be equal.

Step 2: Concept of Probability

Probability is the chance or likelihood that an event will happen. It is measured between 0 and 1 inclusive - where 0 means the event will not happen and 1 means the event is certain to happen. Here, with two possible outcomes, the chance for one outcome is calculated as 1 divided by the number of possible outcomes, which equals 0.5.

Step 3: Applying Probability to the Statement

Assuming no other influences or biases, and considering that there are only two options - Democrat or Republican - the likelihood could be equally split between these two. So, the probability of the next U.S. President being a Republican is indeed 0.5 or 50%, which does make sense.

Key Concepts

Concept of Probability

Probability is a fascinating area of mathematics that deals with the likelihood of events happening. Imagine you have a die for a game; the chance, or probability, of rolling a four is a measure of how likely that event is to occur. Probability values range from 0 to 1.

• A probability of 0 indicates an impossible event, like rolling a seven on a standard die.
• A probability of 1 indicates a certain event, like rolling any number from one to six on that die.
• Probabilities between 0 and 1 represent events that may or may not happen.

Understanding this helps you gauge the chance of an event happening in real-world scenarios, like elections.

Equally Likely Outcomes

When dealing with probability, the concept of equally likely outcomes simplifies calculations. Equally likely outcomes mean each potential outcome of an event has the same chance of occurring. Consider flipping a coin:

• The two possible outcomes, heads or tails, are equally likely.
• Therefore, the probability of landing on heads is 0.5, and the same applies to tails.

In the context of elections, if we assume the only possibilities are a Democrat or a Republican winning, and there are no biases, these outcomes are equally likely.
This assumption is often a starting point in probability exercises, even though real-world events might differ due to additional factors like voting trends and demographic data.

Probability Calculation

Calculating the probability of an event requires understanding the total number of possible outcomes and how frequently a particular outcome occurs. The formula used for probability is:
\[P(E) = \frac{Number \ of \ Favorable \ Outcomes}{Total \ Number \ of \ Possible \ Outcomes}\]
In the exercise, the two outcomes are Democrat or Republican. Hence:

• The number of favorable outcomes for a Republican president is 1 (only one possible Republican winning).
• The total possible outcomes are 2 (either a Democrat or a Republican wins).

Substituting these into the formula gives:\[P(Republican) = \frac{1}{2} = 0.5\]This simple calculation shows how using basic probability allows for easy understanding of potential outcomes in events like elections, given the assumptions.