Problem 42 - Algebra for College Students

Question

If the odds against Smith winning the election are 2 to 5, then what is the probability that Smith wins the election?

Step-by-Step Solution

Answer

The probability that Smith wins is \(\frac{5}{7}\).

Step 1: Understanding Odds

The odds given are 2 to 5 against Smith winning. This means for every 2 unfavorable outcomes (Smith loses), there are 5 favorable outcomes (Smith wins).

Step 2: Convert Odds to Probability

Probability is calculated as the number of favorable outcomes divided by the total number of outcomes (favorable and unfavorable).

Step 3: Calculate Total Outcomes

The total number of outcomes is the sum of favorable and unfavorable outcomes. Here it is 2 (unfavorable) + 5 (favorable) = 7.

Step 4: Determine Probability

The probability of Smith winning is the number of favorable outcomes divided by the total number of outcomes. So, the probability is \(\frac{5}{7}\).

Key Concepts

Understanding Odds

The concept of odds is an essential part of probability. When we talk about odds, we refer to the ratio of unfavorable outcomes to favorable outcomes. In the context of the exercise, the odds against Smith winning the election are given as 2 to 5. This means that for every 2 unfavorable outcomes (situations where Smith does not win), there are 5 favorable outcomes (situations where Smith wins). Odds help us understand how likely an event is to happen in comparison to it not happening.

Favorable Outcomes

Favorable outcomes are the outcomes we are interested in, which represent the event we want to occur. In this exercise, the favorable outcomes are the scenarios where Smith wins the election. Since the odds are 2 to 5 against Smith, we know that there are 5 favorable outcomes. To calculate the probability of a favorable outcome, we consider not just the favorable outcomes, but the total number of outcomes.

Unfavorable Outcomes

Unfavorable outcomes refer to the outcomes we do not want, which represent the event not happening. In this situation, the unfavorable outcomes are the scenarios where Smith loses the election. The given odds are 2 to 5 against Smith, which tells us there are 2 unfavorable outcomes. To find the probability of Smith winning, we must account for both the favorable and unfavorable outcomes.