Problem 4 - Precalculus with Limits

Question

In Exercises 1 - 7, fill in the blanks. If \( P\left(E\right) = 0 \), then \( E \) is an ______ event, and if \( P\left(E\right) = 1 \), then \( E \) is a _______ event.

Step-by-Step Solution

Answer

An event with \(P(E) = 0\) is termed 'impossible' and an event with \(P(E) = 1\) is called 'certain'.

Step 1: Understand probability values

Firstly, one needs to know what it means when an event has a probability of 0 or 1. Probability is a measure that is used to quantify the expectation of an event to occur. It falls within a range between 0 and 1 inclusive.

Step 2: Recognize when an event is certain or impossible

After understanding the range of probability, then one can have the knowledge that an event with a probability of 0 means the event is not going to occur under the given conditions - it's impossible. On the other hand, if the event's probability is 1, it means the event is certain to happen under the given conditions.

Step 3: Fill in the blanks

Having recognized the terms associated with probabilities 0 and 1, the blanks can be filled. The event \(E\) with a probability of \(P(E) = 0\) is called an impossible event, and the event \(E\) with a probability of \(P(E) = 1\) is a certain event.

Key Concepts

Impossible Event

An impossible event is a scenario that has no chance of occurring. It's like trying to roll a 7 on a standard six-sided die. Since the numbers only go up to 6, a 7 is impossible.

In probability, if an event's probability is 0, it means the event cannot happen. The probability scale begins at 0, representing this impossibility.

Consider real-world examples to understand impossible events better.

• A fish walking on land unaided is an impossible event.
• Rolling a number-less side on a standard die is impossible, since each side has a number.

Understanding impossible events helps in setting realistic expectations and differentiating feasible events from wishful thinking.

Certain Event

A certain event is the opposite of an impossible event. It's something guaranteed to happen, like the sun rising in the east.

In terms of probability, an event with a probability of 1 is inevitable. It will occur no matter what. The upper limit of probability is 1, representing certainty.

Real-life examples can help illustrate certain events further:

• Dropping an object and having it fall due to gravity—this is certain.
• Flipping a switch and expecting the light to turn on (in a working setup)—this is usually certain.

Recognizing certain events allows one to plan based on sure outcomes.

Probability Values

Probability values range from 0 to 1, where 0 signifies impossibility and 1 indicates certainty. The values in between express how likely an event is to occur.

For example, a probability of 0.5 means there's a 50/50 chance the event will occur. These values help us make decisions based on likelihood.

Here are key points about probability values:

• They are always between 0 and 1, inclusive.
• A probability closer to 0 suggests a less likely event.
• A probability closer to 1 implies a more likely event.

Understanding these values gives insight into risks and likelihood, allowing better choices to be made.