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 = {rr, rb, br, bb}. let x represent the number of times blue occurs. which of the following is the probability distribution, ( p_x(x) )?

x	( p_x(x) )
1	0.5
2	0.25

x	( p_x(x) )
1	0.33
2	0.33

x	( p_x(x) )
1	0.5
2	0

(note: the last table is partially visible but the first three options are fully presented as above.)

Explanation:

Step1: Determine possible outcomes

The sample space \( S = \{RR, RB, BR, BB\} \), so there are 4 equally - likely outcomes (since the spinner is fair, each spin has probability \( \frac{1}{2} \) for red or blue, and for two spins, by the multiplication rule, each outcome in \( S \) has probability \( \frac{1}{2}\times\frac{1}{2}=\frac{1}{4} \)).

Step2: Calculate probabilities for \( X = 0,1,2 \)

• For \( X = 0 \) (no blue occurrences): The outcome is \( RR \). There is 1 such outcome. So \( P(X = 0)=\frac{\text{Number of outcomes with }X = 0}{\text{Total number of outcomes}}=\frac{1}{4} = 0.25 \).
• For \( X = 1 \) (one blue occurrence): The outcomes are \( RB \) and \( BR \). There are 2 such outcomes. So \( P(X = 1)=\frac{2}{4}=0.5 \).
• For \( X = 2 \) (two blue occurrences): The outcome is \( BB \). There is 1 such outcome. So \( P(X = 2)=\frac{1}{4}=0.25 \).

Answer:

The first table (with \( X = 0 \) having \( P_X(x)=0.25 \), \( X = 1 \) having \( P_X(x) = 0.5 \), and \( X = 2 \) having \( P_X(x)=0.25 \))