Question

PROBLEM SOLVING You and your friend are among several candidates running for class president. You estimate that there is a \(45 \%\) chance you will win and a \(25 \%\) chance your friend will win. What is the probability that you or your friend win the election?

Short answer

The probability that either 'you' or 'your friend' win the election is \(70 \% \).

Step-by-step solution

Understand the Problem

Identify the probabilities given for each candidate winning. The chance that 'you' win is \(45 \%\) and the chance that 'your friend' wins is \(25 \%\). Because 'you' and 'your friend' can't win at the same time, these events are mutually exclusive and the probability that either 'you win' or 'your friend wins' can simply be calculated by adding the individual probabilities.

Convert Percentages to Decimal Form

In calculations with probabilities it's often easier to work with probabilities as decimals rather than percentages. To convert a percentage to a decimal, divide it by 100. Hence, \(45 \% = 0.45\) for 'you' and \(25 \% = 0.25\) for 'your friend'.

Sum the Individual Probabilities

Add the individual probabilities to find the combined probability: \(0.45 (you win) + 0.25 (friend wins) = 0.70\)

Convert Back to Percentage

Use reverse operation to convert the decimal back to a percentage. Multiply the decimal by 100 to get a percentage: \(0.70*100 = 70 \% \)

Key concepts

Understanding Mutually Exclusive Events in Probability

When dealing with probabilities, understanding the concept of mutually exclusive events is crucial. Two events are considered mutually exclusive if they cannot occur at the same time. In our example with the class president election, the events 'you winning' and 'your friend winning' are mutually exclusive since only one candidate can win the election.

This understanding provides the key to calculating probabilities in such scenarios. To find the probability that one of two mutually exclusive events will occur, one simply adds the probabilities of each event happening separately. However, this rule only applies when the events cannot coincide—remember, if events can occur together, they are not mutually exclusive, and different probability rules would apply.

Converting Percentages to Decimals for Probability Calculations

When working with probabilities, it's often more convenient to use decimal forms rather than percentages. The conversion process is straightforward: simply divide the percentage by 100. This means that if you have a probability given as a percentage, like the 45% chance of winning the election, it becomes 0.45 in decimal form.

Converting to decimals is not just a cosmetic change—it simplifies calculations, especially when dealing with multiple probabilities. It's also essential for many statistical calculations and for using probability values in equations and formulas. As a general rule, remember that a percentage is essentially a number out of 100, so dividing by 100 aligns the value with a number out of 1, which is what a probability represents.

Summing Probabilities of Mutually Exclusive Events

Once probabilities of mutually exclusive events have been converted to decimals, summing them is straightforward. To find the combined probability of either event occurring, simply add the decimal probabilities together. Using our election example, you've calculated a 0.45 chance of winning for yourself and a 0.25 chance for your friend. The sum of these, 0.45 + 0.25, gives a 0.70 total probability that either you or your friend will win.

It's important to note, this additive rule only works for mutually exclusive events. If you mistakenly use this approach for events that are not mutually exclusive, you'll end up with incorrect probabilities. To make sure your calculations are correct, always first check whether the events can occur at the same time or not.