Question

a spinner contains four sections: red, blue, green, and yellow. joaquin spins the spinner twice. the set of outcomes is given as $s = \\{rb, rg, ry, rr, br, bg, by, bb, gr, gb, gy, gg, yr, yb, yg, yy\\}$. if the random variable is “yellow (y),” which of the following is the correct probability distribution?

yellow: $x$	probability: $p_x(x)$
1	0.25
2	0

yellow: $x$	probability: $p_x(x)$
1	0.25
2	0.25

yellow: $x$	probability: $p_x(x)$
1	0.375
2	0.125

yellow: $x$	probability: $p_x(x)$
1	0.375
2	0.0625

Explanation:

Step1: Count total outcomes

The sample space \( S \) has \( n(S) = 16 \) outcomes (since \( 4\times4 = 16 \), as we spin twice with 4 sections each time).

Step2: Find \( X = 0 \) (no yellow)

Count outcomes with no Y: For each spin, we can have R, B, G. So for two spins, the number of outcomes is \( 3\times3 = 9 \). Probability \( P(X = 0)=\frac{9}{16}=0.5625 \).

Step3: Find \( X = 1 \) (one yellow)

Number of outcomes with one Y: We can have Y in first spin and non - Y in second, or non - Y in first spin and Y in second. Number of outcomes: \( 3 + 3=6 \). Probability \( P(X = 1)=\frac{6}{16}=0.375 \).

Step4: Find \( X = 2 \) (two yellows)

Number of outcomes with two Ys: \( 1\times1 = 1 \) (YY). Probability \( P(X = 2)=\frac{1}{16}=0.0625 \).

Answer:

The correct probability distribution is the fourth one (the one with \( P(X = 0)=0.5625 \), \( P(X = 1)=0.375 \), \( P(X = 2)=0.0625 \)):

Yellow: \( X \)	Probability: \( P_X(x) \)
1	0.375
2	0.0625