Let's carefully analyze each statement related to the defined event [tex]\( A \)[/tex].
1. The complement of [tex]\(A\)[/tex] is the spinner not landing on blue.
- True. The complement of event [tex]\( A \)[/tex], denoted by [tex]\( A' \)[/tex] or [tex]\(-A\)[/tex], includes all outcomes where the event [tex]\( A \)[/tex] does not occur. Since [tex]\( A \)[/tex] is defined as the spinner landing on blue, [tex]\( A' \)[/tex] would be the spinner landing on any color other than blue (i.e., red, yellow, or green).
2. [tex]\( P(A) = \frac{1}{2} \)[/tex]
- False. There are four equal-sized sections, and each is equally likely to occur. Since the spinner has four sections (red, yellow, blue, and green), the probability of landing on any one section, including blue, is [tex]\( \frac{1}{4} \)[/tex], not [tex]\( \frac{1}{2} \)[/tex].
3. There are three favorable outcomes of [tex]\(-A\)[/tex].
- True. The event [tex]\(-A\)[/tex] or [tex]\(A'\)[/tex] corresponds to the spinner not landing on blue. The favorable outcomes for [tex]\(-A\)[/tex] are the sections red, yellow, and green—three outcomes in total.
4. [tex]\( P(-A) = \frac{3}{4} \)[/tex]
- True. Since the probability of landing on blue ([tex]\( P(A) \)[/tex]) is [tex]\( \frac{1}{4} \)[/tex], the probability of the complement event [tex]\(-A\)[/tex] (not landing on blue) is [tex]\( 1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4} \)[/tex].
5. It is certain that either [tex]\( A \)[/tex] or [tex]\(-A\)[/tex] will occur.
- True. By definition of probability, the event [tex]\( A \)[/tex] (landing on blue) and the event [tex]\(-A\)[/tex] (not landing on blue) cover all possible outcomes of the spinner. Therefore, it is certain that one of these two events will occur, which is the essence of a complement.
Therefore, the true statements are:
- The complement of [tex]\( A \)[/tex] is the spinner not landing on blue.
- There are three favorable outcomes of [tex]\(-A\)[/tex].
- [tex]\( P(-A) = \frac{3}{4} \)[/tex].
- It is certain that either [tex]\( A \)[/tex] or [tex]\(-A\)[/tex] will occur.
