Probability models: Equally likely outcomes PROBLEM: Suppose you spin a spinner with three equal sections (red, blue, and yellow) two times. (a) Give a probability model for this chance process: (b)Define event A as spinning blue at least once. Find $$ P(A) $$.

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## **Problem:**

Suppose you spin a spinner with three equal sections (Red, Blue, and Yellow) two times.

### **(a) Give a probability model for this chance process:**

**Solution:**

A **probability model** consists of all possible outcomes of a chance process along with their associated probabilities.

1. **Identify the Sample Space (S):**
   
   Since the spinner has three equally likely sections—Red (R), Blue (B), and Yellow (Y)—each spin can result in one of these three outcomes. When spinning the spinner **twice**, the outcomes are ordered pairs representing the results of each spin.

$$
   S = \{ (R, R),\ (R, B),\ (R, Y),\ (B, R),\ (B, B),\ (B, Y),\ (Y, R),\ (Y, B),\ (Y, Y) \}
   $$

2. **Determine the Probability of Each Outcome:**
   
   - Each section is equally likely, so the probability of landing on any one color in a single spin is $$\frac{1}{3}$$.
   - Since the two spins are independent, the probability of any ordered pair is the product of the probabilities of each individual spin.

$$
   P((X, Y)) = P(X) \times P(Y) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}
   $$

3. **Probability Model:**

| Outcome | Probability |
   |---------|-------------|
   | (R, R)  | $$ \frac{1}{9} $$ |
   | (R, B)  | $$ \frac{1}{9} $$ |
   | (R, Y)  | $$ \frac{1}{9} $$ |
   | (B, R)  | $$ \frac{1}{9} $$ |
   | (B, B)  | $$ \frac{1}{9} $$ |
   | (B, Y)  | $$ \frac{1}{9} $$ |
   | (Y, R)  | $$ \frac{1}{9} $$ |
   | (Y, B)  | $$ \frac{1}{9} $$ |
   | (Y, Y)  | $$ \frac{1}{9} $$ |

### **(b) Define event A as spinning blue at least once. Find $$ P(A) $$.**

**Solution:**

**Event A** occurs if at least one of the two spins results in Blue (B). To find $$ P(A) $$, we can use one of two methods:

1. **Counting Favorable Outcomes:**

First, identify all outcomes in the sample space where at least one spin is Blue.

$$
   A = \{ (R, B),\ (B, R),\ (B, B),\ (B, Y),\ (Y, B) \}
   $$

There are **5** favorable outcomes.

$$
   P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{5}{9}
   $$

2. **Using the Complement Rule:**

Alternatively, it's often easier to calculate the probability of the complementary event (event $$ A' $$) where **no** spins result in Blue, and then subtract from 1.

- **Event $$ A' $$:** Neither spin is Blue.

The outcomes where neither spin is Blue are:

$$
   A' = \{ (R, R),\ (R, Y),\ (Y, R),\ (Y, Y) \}
   $$

There are **4** such outcomes.

$$
   P(A') = \frac{4}{9}
   $$

Therefore,

$$
   P(A) = 1 - P(A') = 1 - \frac{4}{9} = \frac{5}{9}
   $$

**Final Answer:**

### **(b) $$ P(A) = \frac{5}{9} $$**
**(a) Probability Model:** - Total outcomes: 9 - Each outcome has a probability of $$ \frac{1}{9} $$. **(b) Probability of Event A ($$ P(A) $$):** $$ \frac{5}{9} $$

Probability models: Equally likely outcomes PROBLEM: Suppose you spin a spinner with three equal sections (red, blue, and yellow) two times. (a) Give a probability model for this chance process: (b)Define event A as spinning blue at least once. Find \( P(A) \).

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