Question

a spinner is divided into two equal parts, one red and one blue. the set of possible outcomes when the spinner is spun twice is \\(s = \\{\text{rr}, \text{rb}, \text{br}, \text{bb}\\}\\). let \\(x\\) represent the number of times blue occurs. which of the following is the probability distribution, \\(p_x(x)\\)?

Explanation:

Identify the sample space and outcomes

Using the Sample Space knowledge point
The spinner has two equal parts: Red (R) and Blue (B). Since the parts are equal, the probability of landing on Red is \(0.5\) and on Blue is \(0.5\).
When spun twice, the sample space of equally likely outcomes is:

$$ S = \{RR, RB, BR, BB\} $$

Each of these 4 outcomes has a probability of:

$$ P(\text{outcome}) = 0.5 \times 0.5 = 0.25 $$

Define the random variable X

The random variable \(X\) represents the number of times blue occurs in two spins. We evaluate \(X\) for each outcome in the sample space:

• For \(RR\): \(X = 0\) (no blue)
• For \(RB\): \(X = 1\) (one blue)
• For \(BR\): \(X = 1\) (one blue)
• For \(BB\): \(X = 2\) (two blues)

Calculate the probability distribution

Using the Probability Distribution knowledge point
We find the probability for each possible value of \(X\):

• For \(X = 0\):

$$ P_X(0) = P(\{RR\}) = 0.25 $$

• For \(X = 1\):

$$ P_X(1) = P(\{RB, BR\}) = 0.25 + 0.25 = 0.5 $$

• For \(X = 2\):

$$ P_X(2) = P(\{BB\}) = 0.25 $$

Match with the given options

The calculated probability distribution is:

• \(X = 0 \implies P_X(0) = 0.25\)
• \(X = 1 \implies P_X(1) = 0.5\)
• \(X = 2 \implies P_X(2) = 0.25\)

This matches the first table option.

Answer:

• **(A)

$$\begin{array}{|c|c|} \hline X & P_X(x) \\ \hline 0 & 0.25 \\ \hline 1 & 0.5 \\ \hline 2 & 0.25 \\ \hline \end{array}$$

(Correct answer)**

• (B)

$$\begin{array}{|c|c|} \hline X & P_X(x) \\ \hline 0 & 0.33 \\ \hline 1 & 0.33 \\ \hline 2 & 0.33 \\ \hline \end{array}$$

• (C)

$$\begin{array}{|c|c|} \hline X & P_X(x) \\ \hline 0 & 0.5 \\ \hline 1 & 0.5 \\ \hline 2 & 0 \\ \hline \end{array}$$

• (D)

$$\begin{array}{|c|c|} \hline X & P_X(x) \\ \hline 0 & 0 \\ \hline 1 & 0.5 \\ \hline 2 & 0.5 \\ \hline \end{array}$$