Problem 39 - Precalculus with Limits

Question

In Exercises 39 - 42, you are given the probability that an event will not happen. Find the probability that the event will happen. \( P(E') = 0.23 \)

Step-by-Step Solution

Answer

The probability that the event E will happen is 0.77.

Step 1: Understanding the problem

We know that the sum of the probabilities of an event E and its complement \( E' \) (the event not happening) is always 1. This is based on the fact that either the event happens or it does not. Therefore, the formula to calculate the probability that event E will happen, given the probability of it not happening, is \( P(E) = 1 - P(E') \)

Step 2: Apply the formula to find the probability of E happening

We substitute the given value of \( P(E') = 0.23 \) into the formula from the previous step: \( P(E) = 1 - P(E') = 1 - 0.23 = 0.77 \).

Key Concepts

Complementary Events

In probability theory, understanding complementary events is essential. These are pairs of outcomes where one outcome always happens, and the other does not. Think of it like flipping a coin: if the coin lands on heads, it can't also land on tails. This means heads and tails are complementary events.

For any event, say \( E \), the complement of that event, \( E' \), is the event that \( E \) does not occur.

• The probability of the event and its complement will always add up to 1, meaning something will definitely happen.
• In mathematical terms: \( P(E) + P(E') = 1 \).

This concept is foundational because it helps to simplify complex probability calculations by breaking them down into more manageable parts.

Probability Formula

The probability formula is a key tool for calculating how likely an event is to occur. Probability is a measure of uncertainty and is expressed as a number between 0 and 1.

To find the probability of an event \( E \), especially when you know the probability that it does not occur (the complement), use this simple formula:

• \( P(E) = 1 - P(E') \)
• This formula is handy because it lets you find the likelihood of an event occurring if you know how often it doesn't happen.
• For example, if the probability of rain not occurring on a particular day is 0.23, you can easily find the probability of rain by subtracting 0.23 from 1, which is 0.77.

Utilizing this formula can help simplify problems, keeping calculations straightforward.

Probability Calculation

Calculating probabilities can feel tricky at first, but it's all about following simple steps. When you're given the probability of an event not occurring and need to find out how likely it is to happen, here's how you do it:

• Start with the formula: \( P(E) = 1 - P(E') \).
• Substitute the known value into the formula. In our example, use \( P(E') = 0.23 \).
• Calculate: \( P(E) = 1 - 0.23 = 0.77 \).

This calculation tells you that there's a 77% chance the event will happen.

Breaking it down into steps like this makes it easier to handle different probability questions, while reinforcing your understanding of complementary events. The repetition of this process helps solidify the fundamental concepts of probability in your mind.