Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

You are given the probability that an event will not happen. Find the probability that the event will happen. $$P\left(E^{\prime}\right)=\frac{61}{100}$$

Step-by-Step Solution

Answer

The probability that the event will happen is \(\frac{39}{100}\).

Step 1: Understanding complementary events

In probability, an event E and its complementary event \(E^{\prime}\) cover all possible outcomes. Therefore, the sum of their probabilities equals 1. Using this property, the formula to calculate \(P(E)\) is: \(P(E) = 1 - P(E^{\prime})\)

Step 2: Apply the formula

From the exercise, we know that \(P(E^{\prime})=\frac{61}{100}\). Substitute this into the formula from Step 1 to find \(P(E)\): \(P(E) = 1 - \frac{61}{100}\)

Step 3: Simplify

Subtract \(\frac{61}{100}\) from 1 to find the probability \(P(E)\): \(P(E) = \frac{39}{100}\)

Key Concepts

Complementary Events

Whenever we deal with probabilities, it is essential to understand the concept of complementary events. A complementary event refers to the opposite of another event. In simple terms, if event \( E \) is one possible result in an experiment, then the complementary event \( E^{\prime} \) includes all other outcomes that are not \( E \). Together, \( E \) and \( E^{\prime} \) represent all possible outcomes of a given scenario.
This means their probabilities add up to 1. For example, if you know the probability of a coin landing on heads is \( 0.5 \), the probability of it landing on tails (the complement) is also \( 0.5 \).
This relationship is crucial because by understanding an event’s complement, you can easily calculate the likelihood of the event itself occurring. It's like understanding that the probability of not rolling a six on a die (5 out of 6 outcomes) helps you find the probability of rolling a six. If the complement probability is given, subtract it from 1 to get the event's probability.

Probability Formula

Calculating probabilities becomes easier with a solid grasp of probability formulas. For complementary events, the basic formula is:
\[ P(E) = 1 - P(E^{\prime}) \] This equation states that the probability of event \( E \) happening is equal to one minus the probability of its complement \( E^{\prime} \) occurring.
In any given experiment, if you’re provided with the probability of \( E^{\prime} \), you can quickly find \( P(E) \) using this straightforward relationship. For instance, if \( P\left(E^{\prime}\right) = \frac{61}{100} \), plugging it into the formula yields:
\[ P(E) = 1 - \frac{61}{100} = \frac{39}{100} \] This tells you that the likelihood of event \( E \) occurring is \( \frac{39}{100} \) or 39%. Such formulas not only simplify calculations but also reinforce the conceptual understanding of probability problems.

Event Occurrence Likelihood

Determining the likelihood of an event occurring is a fundamental aspect of probability. It is the process of assessing how probable it is for a certain event to take place. This is expressed as a number between 0 and 1, where 0 means the event will never occur, and 1 indicates certainty of occurrence.
Different methods address this probability depending on the data or scenario available, such as historical data analysis or theoretical reasoning. In our example exercise, using the complement’s probability helped us to find that the likelihood of the event \( E \) occurring is \( \frac{39}{100} \).
Understanding the likelihood plays a significant role in decision-making, from predicting weather patterns to strategically planning in games. Whether an event is rare or common can determine strategies and choices in both everyday contexts and professional fields.