Finding the Probability of an Event In Exercises 21-24, the proba...

4. Probability

Complements

Statistics

4. Probability

Complements

1:14 minutes

Problem 3.1.21

Textbook Question

Finding the Probability of an Event In Exercises 21-24, the probability that an event will not happen is given. Find the probability that the event will happen.
21. P(E') =0.95

Verified step by step guidance

Step 1: Understand the problem. The probability of an event not happening, denoted as P(E'), is given as 0.95. You are tasked with finding the probability of the event happening, denoted as P(E).

Step 2: Recall the relationship between the probability of an event happening and not happening. The sum of these probabilities must equal 1, as they are complementary events. Mathematically, this is expressed as:

P(E) + P(E') = 1

Step 3: Rearrange the formula to solve for P(E). Subtract P(E') from both sides:

P(E) = 1 - P(E')

Step 4: Substitute the given value of P(E') into the formula. In this case, P(E') = 0.95, so the equation becomes:

P(E) = 1 - 0.95

Step 5: Perform the subtraction to find the probability of the event happening. This will give you the value of P(E).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complementary Events

In probability theory, complementary events are two outcomes of a single event that cannot occur at the same time. If an event E occurs, its complement E' (not E) represents all outcomes where E does not happen. The sum of the probabilities of an event and its complement is always equal to 1, which is a fundamental principle in probability.

Probability Calculation

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. To find the probability of an event happening, you can subtract the probability of the event not happening from 1. For example, if P(E') = 0.95, then P(E) = 1 - P(E') = 0.05, indicating a low likelihood of the event occurring.

Basic Probability Rules

Basic probability rules include the addition and multiplication rules, which help in calculating the probabilities of combined events. The addition rule states that the probability of either of two mutually exclusive events occurring is the sum of their individual probabilities. Understanding these rules is essential for solving more complex probability problems and for interpreting results correctly.

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Finding the Probability of an Event In Exercises 21-24, the probability that an event will not happen is given. Find the probability that the event will happen. 21. P(E') =0.95