Question

the fair spinner shown is spun 1 time. there are four possible outcomes, one for each color. based on the picture, which statements are true? the probability of each outcome is equal the probability of each outcome is different the outcomes have a uniform probability distribution the outcomes have a nonuniform distribution what is the probability of spinning a yellow? complete, 1/4 what is the probability of spinning a yellow or a green? complete, 0.5 what is the sum of the probabilities for all possible outcomes? retry, box with 1?

Explanation:

1. Probability of spinning yellow

Step1: Identify the spinner's sections

The fair spinner has 4 equal - sized sections (one for each color: yellow, blue, red, green).

Step2: Calculate the probability

For a fair spinner with $n$ equal - sized sections, the probability of landing on a specific section is $\frac{1}{n}$. Here, $n = 4$ and we want the probability of landing on yellow. So the probability $P(\text{yellow})=\frac{1}{4}$.

2. Probability of spinning yellow or green

Step1: Find individual probabilities

We know from the first part that $P(\text{yellow})=\frac{1}{4}$ and since the spinner is fair, $P(\text{green})=\frac{1}{4}$ (because there are 4 equal sections).

Step2: Use the addition rule for mutually exclusive events

For mutually exclusive events (you can't land on yellow and green at the same time), $P(A\cup B)=P(A)+P(B)$. So $P(\text{yellow or green})=P(\text{yellow}) + P(\text{green})=\frac{1}{4}+\frac{1}{4}=\frac{2}{4}=0.5$.

3. Sum of probabilities for all possible outcomes

Step1: Recall the probability axiom

One of the fundamental axioms of probability is that the sum of the probabilities of all possible outcomes in a sample space is equal to 1.

Step2: Apply to the spinner

The possible outcomes are yellow, blue, red, and green. Let their probabilities be $P(\text{yellow})$, $P(\text{blue})$, $P(\text{red})$, and $P(\text{green})$. Since the spinner is fair, each has a probability of $\frac{1}{4}$. So the sum is $\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{4}{4}=1$.

4. True statement about the spinner's probability distribution

Answer:

$\frac{1}{4}$