Problem 38 - Precalculus with Limits

Question

In Exercises 35 - 38, you are given the probability that an event will happen. Find the probability that the event will not happen. \( P(E) = \dfrac{2}{3} \)

Step-by-Step Solution

Answer

The probability that the event E will not happen is \( \dfrac{1}{3} \)

Step 1: Identify the given probability

The probability of event E, denoted as \(P(E)\), is given as \( \dfrac{2}{3} \)

Step 2: Calculate the complementary probability

Since the total probability equals 1, the probability of the complementary event \( P(E') \) or \( P(E^c) \) can be found by subtracting the given probability from 1. This gives us \( P(E') = 1 - P(E) \). Substituting the given probability we obtain \( P(E') = 1 - \dfrac{2}{3} \)

Step 3: Simplify the result

The exact arithmetic between \( 1 \) and \( \dfrac{2}{3} \) results in \( P(E') = \dfrac{1}{3} \). Therefore, the probability of the event E not occurring is \(\dfrac{1}{3}\).

Key Concepts

Complementary probability

Complementary probability is an essential concept in probability theory, helping us understand the likelihood of an event not occurring. Let's say we have an event, denoted as event E, with its probability given as \( P(E) \). Complementary probability, noted as \( P(E') \) or \( P(E^c) \), represents the chance of this event not taking place.

The key relationship to remember is that the total probability of all possible outcomes of an event and its complement is always 1. Thus, to find the complementary probability, we simply subtract the event's probability from 1:

• Formula: \( P(E') = 1 - P(E) \).

For instance, if the probability of an event occurring is \( \frac{2}{3} \), you can find the complement by doing some simple subtraction:

• \( P(E') = 1 - \frac{2}{3} = \frac{1}{3} \).

This tells us that there is a \( \frac{1}{3} \) chance that this event will not occur.

Event probability

Event probability is the likelihood of a specific outcome or event occurring when we perform a random experiment. In probability theory, this is represented as \( P(E) \), where E is the event in question. The value of \( P(E) \) is between 0 and 1, with 0 indicating impossibility and 1 indicating certainty.

Here are some important points to remember:

• An event with \( P(E) = 0 \) cannot happen.
• An event with \( P(E) = 1 \) will undoubtedly happen.
• Probabilities between 0 and 1 represent varying levels of likelihood.

Calculating event probability involves analyzing all possible outcomes. For example, if drawing a colored ball from a bag, the event probability might consider how many of each color are present. If the given probability \( P(E) \) for drawing a particular colored ball is \( \frac{2}{3} \), it means out of three possible draws, two are expected to be successful if repeated over a large number of tries.

Arithmetic operations in probability

Arithmetic operations in probability, such as addition and subtraction, provide a means to calculate more complex probabilities from basic ones. One common situation involves finding the complementary probability, as seen in our example.

To determine the probability of an event not happening, you use subtraction:

• Subtract the event's probability from 1 to get the complementary probability: \( P(E') = 1 - P(E) \).

Moreover, addition can help when calculating the probability of either of two non-overlapping events occurring, known as mutually exclusive events, using:

• \( P(A \text{ or } B) = P(A) + P(B) \).

Understanding how to efficiently use these basic arithmetic principles can greatly enhance your ability to solve probability problems efficiently. They are particularly useful when dealing with more complex situations involving multiple events.