Problem 51 - Precalculus with Limits

Question

Three people have been nominated for president of a class. From a poll, it is estimated that the first candidate has a \( 37\% \) chance of winning and the second candidate has a \( 44\% \) chance of winning. What is the probability that the third candidate will win?

Step-by-Step Solution

Answer

The probability that the third candidate will win is \( 0.19 \) or \( 19\% \)

Step 1: Identify given probabilities

Candidate 1 has a \( 37\% \) chance of winning and Candidate 2 has a \( 44\% \) chance of winning.

Step 2: Convert the percentages to decimals

Converting the chances of winning to decimals, we have Candidate 1 with a chance of winning equal to \( 0.37 \) and Candidate 2 equal to \( 0.44 \)

Step 3: Sum the probabilities of the first two candidates

The combined probability of both Candidate 1 and 2 winning is \( 0.37 + 0.44 = 0.81 \)

Step 4: Find the probability of the third candidate winning

We then subtract the combined probability from 1 to find the probability that Candidate 3 will win the election, that is \( 1 - 0.81 = 0.19 \). This means the third candidate has a \( 19\% \) chance of winning.

Key Concepts

Percentages

Percentages are a way of expressing a number as a fraction of 100. They are often used to compare ratios or proportions. In the context of probability, percentages help quantify the likelihood of an event occurring. For instance, if a candidate has a 37% chance of winning, this means out of every 100 scenarios, approximately 37 will result in them winning. To convert a percentage to a decimal, which is often essential for calculations, you simply divide by 100. So, 37% becomes 0.37. This is useful because many mathematical operations with percentages involve multiplication or addition, which are easier to perform with decimals. Percentages give a straightforward way to communicate probabilities to a wide audience.

Probability Calculation

Probability calculation helps us quantify the likelihood of different outcomes. In our case, we are dealing with multiple events, each of which is mutually exclusive. This means Candidate 1, Candidate 2, and Candidate 3 cannot win simultaneously. A fundamental rule of probability is that the sum of probabilities for all possible outcomes equals 1 (or 100%).

• First, we convert the percentages to decimals: Candidate 1 (0.37), Candidate 2 (0.44).
• Next, add these probabilities to find their total (0.81).
• Finally, subtract this sum from 1 to find the probability for Candidate 3 (1 - 0.81 = 0.19).

Thus, the probability that Candidate 3 will win is 0.19, or 19%. In summary, probability calculations allow us to allocate "chances" to each possibility and ensure our overall accounting is correct.

Problem-Solving Steps

When faced with a probability problem, a structured approach is key to finding the solution. Let's break it down into manageable steps using our candidate example. - **Identify Given Information:** Start by listing known probabilities. Here, Candidate 1 has a 37% chance and Candidate 2 has a 44% chance. - **Convert Percentages:** Transform these percentages into decimal form for easier computation (0.37 and 0.44, respectively). - **Calculate Known Outcomes:** Add known probabilities to check their total (0.81). - **Find Unknown Probability:** Subtract this total from 1 to find the probability of the remaining outcome. In our exercise, subtracting from 1 gives the third candidate a 19% chance of winning. These steps help in methodically tackling probability problems by narrowing down unknowns, simplifying numbers, and ensuring each outcome is accounted for. Always verify your results to ensure they make logical sense within the context of the problem.