Finding the Probability of the Complement of an Event In Exercise...

4. Probability

Complements

Statistics

4. Probability

Complements

1:29 minutes

Problem 3.1.19

Textbook Question

Finding the Probability of the Complement of an Event In Exercises 17-20, the probability that an event will happen is given. Find the probability that the event will not happen.
19. P(E)=0.03

Verified step by step guidance

Step 1: Understand the concept of the complement of an event. The complement of an event E, denoted as E', represents all outcomes in the sample space that are not part of event E. The sum of the probabilities of an event and its complement is always equal to 1.

Step 2: Recall the formula for the probability of the complement of an event: \( P(E') = 1 - P(E) \). This formula is derived from the fact that \( P(E) + P(E') = 1 \).

Step 3: Identify the given probability of the event \( P(E) \). In this problem, \( P(E) = 0.03 \).

Step 4: Substitute the given value of \( P(E) \) into the formula for \( P(E') \): \( P(E') = 1 - 0.03 \).

Step 5: Simplify the expression to find \( P(E') \). This will give you the probability that the event will not happen.

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Key Concepts

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

Probability of an Event

Probability quantifies the likelihood of an event occurring, expressed as a number between 0 and 1. A probability of 0 indicates impossibility, while a probability of 1 indicates certainty. In this case, P(E) = 0.03 means there is a 3% chance that event E will occur.

Complement of an Event

The complement of an event refers to all outcomes in which the event does not occur. It is denoted as P(E') and can be calculated using the formula P(E') = 1 - P(E). This concept is essential for determining the probability of an event not happening based on the probability of it happening.

Calculating Complement Probability

To find the probability of the complement of an event, subtract the probability of the event from 1. For example, if P(E) = 0.03, then P(E') = 1 - 0.03 = 0.97. This means there is a 97% chance that event E will not occur, illustrating how to effectively use the complement rule in probability.

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Finding the Probability of the Complement of an Event In Exercises 17-20, the probability that an event will happen is given. Find the probability that the event will not happen.19. P(E)=0.03