Let's break down each statement and verify its validity step by step:
1. The complement of [tex]\( A \)[/tex] is the spinner not landing on blue.
- Event [tex]\( A \)[/tex] is defined as the spinner landing on blue.
- The complement of [tex]\( A \)[/tex] ([tex]\( -A \)[/tex]) would therefore be any outcome where the spinner does not land on blue.
- Since the sections are red, yellow, blue, and green, not landing on blue means landing on either red, yellow, or green.
- This statement is true.
2. [tex]\( P(A) = \frac{1}{3} \)[/tex]
- The probability of event [tex]\( A \)[/tex] (spinner landing on blue) is calculated as the number of favorable outcomes for landing on blue divided by the total number of possible outcomes.
- There is 1 blue section out of 4, so [tex]\( P(A) = \frac{1}{4} \)[/tex].
- [tex]\( P(A) = \frac{1}{4} \neq \frac{1}{3} \)[/tex]
- This statement is false.
3. There are three favorable outcomes of [tex]\( \sim A \)[/tex].
- The complement of [tex]\( A \)[/tex] is the spinner not landing on blue.
- The favorable outcomes for [tex]\( -A \)[/tex] are landing on red, yellow, or green.
- Hence, there are 3 favorable outcomes for [tex]\( -A \)[/tex].
- This statement is true.
4. [tex]\( P(-A) = \frac{3}{4} \)[/tex]
- The probability of the complement of event [tex]\( A \)[/tex] ([tex]\( -A \)[/tex]) is calculated as the number of outcomes where the spinner does not land on blue divided by the total number of possible outcomes.
- There are 3 non-blue sections out of 4.
- Thus, [tex]\( P(-A) = \frac{3}{4} \)[/tex].
- This statement is true.
5. It is certain that either [tex]\( A \)[/tex] or [tex]\( -A \)[/tex] will occur.
- In probability theory, an event and its complement are mutually exclusive and together they cover all possible outcomes.
- Thus, either the spinner lands on blue ([tex]\( A \)[/tex]) or it does not land on blue ([tex]\( -A \)[/tex]).
- This fact ensures that either [tex]\( A \)[/tex] or [tex]\( -A \)[/tex] must occur.
- This statement is true.
So, the correct answers are:
- The complement of [tex]\( A \)[/tex] is the spinner not landing on blue.
- There are three favorable outcomes of [tex]\( \sim A \)[/tex].
- [tex]\( P(-A) = \frac{3}{4} \)[/tex]
- It is certain that either [tex]\( A \)[/tex] or [tex]\( -A \)[/tex] will occur.
These four statements are true.
