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, $p_x(x)$? four tables with x and $p_x(x)$ values are shown: table 1: x=0 (0.33), x=1 (0.33), x=2 (0.33); table 2: x=0 (0.25), x=1 (0.5), x=2 (0.25); table 3: x=0 (0), x=1 (0.5), x=2 (0.5); table 4: x=0 (0.5), x=1 (0.5), x=2 (0)

Explanation:

Step1: Determine possible \( X \) values

The spinner is spun twice, so \( X \) (number of blue occurrences) can be 0, 1, or 2. The sample space \( S = \{RR, RB, BR, BB\} \), with 4 outcomes.

Step2: Calculate \( P_X(0) \) (blue occurs 0 times)

Outcome with 0 blue: \( RR \). Number of such outcomes: 1.
Probability: \( P_X(0) = \frac{\text{Number of } RR}{\text{Total outcomes}} = \frac{1}{4} = 0.25 \).

Step3: Calculate \( P_X(1) \) (blue occurs 1 time)

Outcomes with 1 blue: \( RB, BR \). Number of such outcomes: 2.
Probability: \( P_X(1) = \frac{\text{Number of } RB, BR}{\text{Total outcomes}} = \frac{2}{4} = 0.5 \).

Step4: Calculate \( P_X(2) \) (blue occurs 2 times)

Outcome with 2 blue: \( BB \). Number of such outcomes: 1.
Probability: \( P_X(2) = \frac{\text{Number of } BB}{\text{Total outcomes}} = \frac{1}{4} = 0.25 \).

Answer:

The probability distribution table with \( X = 0 \) (0.25), \( X = 1 \) (0.5), \( X = 2 \) (0.25) (the second table among the four options).