Question

properties of probability distributions
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) )?
(charts showing different probability distributions for ( x = 0, 1, 2 ) with corresponding ( p_x(x) ) values:

• first chart: ( x=0 ) with ( p_x(x)=0 ), ( x=1 ) with ( p_x(x)=0.5 ), ( x=2 ) with ( p_x(x)=0.5 )
• second chart: ( x=0,1,2 ) each with ( p_x(x)=0.33 )
• third chart: ( x=0 ) with ( p_x(x)=0.25 ), ( x=1 ) with ( p_x(x)=0.5 ), ( x=2 ) with ( p_x(x)=0.25 )
• fourth chart: ( x=0 ) with ( p_x(x)=0.5 ), ( x=1 ) with ( p_x(x)=0.5 ), ( x=2 ) with ( p_x(x)=0 ))

Explanation:

Step1: Analyze possible outcomes

The spinner has 2 equal parts (red, blue). Spun twice, outcomes: \( S = \{RR, RB, BR, BB\} \). \( X \) = number of blue occurrences. So \( X = 0 \) (RR), \( X = 1 \) (RB, BR), \( X = 2 \) (BB).

Step2: Calculate probabilities

• \( P(X = 0) \): Probability of RR. Since each spin is independent, \( P(R) = 0.5 \), so \( P(RR) = 0.5 \times 0.5 = 0.25 \)? Wait, no: Wait, \( P(R) = 0.5 \), \( P(B) = 0.5 \).
• \( X = 0 \): RR. Probability: \( 0.5 \times 0.5 = 0.25 \)? Wait, no, wait the number of outcomes: For \( X = 0 \), only 1 outcome (RR) out of 4? Wait no, the set \( S \) has 4 outcomes: RR, RB, BR, BB. So:
• \( X = 0 \): RR → 1 outcome. Probability: \( \frac{1}{4} = 0.25 \)
• \( X = 1 \): RB, BR → 2 outcomes. Probability: \( \frac{2}{4} = 0.5 \)
• \( X = 2 \): BB → 1 outcome. Probability: \( \frac{1}{4} = 0.25 \)

Step3: Match with tables

Check each table:

• Top - right: \( P(0)=0.33, P(1)=0.33, P(2)=0.33 \) → sums to ~1, but probabilities don't match.
• Top - left: \( P(0)=0, P(1)=0.5, P(2)=0.5 \) → wrong.
• Middle: \( P(0)=0.25, P(1)=0.5, P(2)=0.25 \) → matches our calculation.
• Bottom: \( P(0)=0.5, P(1)=0.5, P(2)=0 \) → wrong.

Answer:

The table with \( P_X(0) = 0.25 \), \( P_X(1) = 0.5 \), \( P_X(2) = 0.25 \) (the middle table among the four, or the third table in the image order: top - left, top - right, middle, bottom).