How is the probability of an event not occurring related to the probability that it does​ occur? Why?

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Part 1

Choose the correct answer below.

A.

Upper P left parenthesis not Upper A right parenthesis equals StartFraction 1 Over Upper P left parenthesis Upper A right parenthesis EndFractionP(not A)=1P(A)

because the product of the probabilities must be 1.

B.

Upper P left parenthesis not Upper A right parenthesis equals 1 minus Upper P left parenthesis Upper A right parenthesisP(not A)=1−P(A)

because the sum of the probabilities must be 1.

C.

Upper P left parenthesis not Upper A right parenthesis equals negative Upper P left parenthesis Upper A right parenthesisP(not A)=−P(A)

because the probabilities have opposite values.

D.

Upper P left parenthesis not Upper A right parenthesis equals Upper P left parenthesis Upper A right parenthesis minus 1P(not A)=P(A)−1

because the probabilities have opposite values.

The correct answer is:

**B. \( P(\text{not } A) = 1 - P(A) \)** because the sum of the probabilities must be 1.

### Explanation:
The probability of an event \( A \) occurring, denoted by \( P(A) \), and the probability of the event not occurring, denoted by \( P(\text{not } A) \), are complementary. This means that one of the two possibilities must happen. Therefore, their probabilities add up to 1, as:

\[
P(A) + P(\text{not } A) = 1
\]

Solving for \( P(\text{not } A) \), we get:

\[
P(\text{not } A) = 1 - P(A)
\]

Thus, the probability of an event not occurring is simply 1 minus the probability of it occurring.

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### Related questions:
1. What is the probability of the complement of an event when the event has a probability of 0.6?
2. If the probability of an event occurring is 0.25, what is the probability of the event not occurring?
3. How do complementary probabilities apply in real-life situations?
4. Can the probability of an event and its complement both be greater than 0.5? Why or why not?
5. How would the probability of not rolling a 6 on a fair die be calculated?

### Tip:
Always remember that the sum of the probabilities of an event and its complement must equal 1!

How is the probability of an event not occurring related to the probability that it does occur? Why?

Explore how the probability of an event not occurring is related to the probability of it occurring using the Complement Rule. Learn about key concepts in probability theory and formulas such as P(not A) = 1 - P(A). Suitable for students in Grades 6-8.

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