20. A box contains two balls, a blue ball and a red ball. Suppose you draw a ball at random from the box and then toss a die. Draw a tree diagram and glve the sample space. a. What is the probability of drawing a blue ball and rolling a 2 ? b. What is the probability of rolling a 3? c. What is the probability of drawing a blue ball? d. What is the probability of drawing a red ball or rolling an even number?

We begin by describing the experiment step by step: Step 1: Drawing a ball from a box containing one blue ball (B) and one red ball (R). Each ball is equally likely, so   P(B) = 1/2 and P(R) = 1/2. Step 2: Tossing a fair six-sided die. Each die roll (numbers 1 through 6) has probability 1/6. Since these actions are independent, the combined outcome probability is the product of the individual probabilities:   1/2 × 1/6 = 1/12 for each outcome. ───────────────────────────── a. Drawing a blue ball and rolling a 2 This event corresponds to the single outcome (B, 2). Its probability is:   P(B and 2) = 1/12. ───────────────────────────── b. Rolling a 3 The event “rolling a 3” can occur whether you drew a blue ball or a red ball. Hence, the favorable outcomes are:   (B,3) and (R,3). Probability = 1/12 + 1/12 = 2/12 = 1/6. ───────────────────────────── c. Drawing a blue ball Regardless of the die, drawing a blue ball happens in the first step. Since the die roll does not affect the ball drawing, we have:   P(B) = 1/2. ───────────────────────────── d. Drawing a red ball or rolling an even number First, list the outcomes for each event: 1. Red ball outcomes:   (R,1), (R,2), (R,3), (R,4), (R,5), (R,6)  (total = 6 outcomes). 2. Even number outcomes when a blue ball is drawn:   (B,2), (B,4), (B,6)  (total = 3 outcomes). Notice that if you roll an even number after drawing a red ball, it’s already part of the red ball outcomes. Hence, we should only add the blue-even outcomes to avoid double counting. Total favorable outcomes = 6 (red ball outcomes) + 3 (blue ball even outcomes) = 9. Probability = Number of favorable outcomes / Total outcomes = 9/12 = 3/4. ───────────────────────────── Tree Diagram and Sample Space A tree diagram helps illustrate all possible outcomes:             [B] (Probability = 1/2)                [R] (Probability = 1/2)                / | \                      / | \         Die outcomes: 1, 2, 3, 4, 5, 6        Die outcomes: 1, 2, 3, 4, 5, 6 Each branch (for every ball and die combination) represents an outcome of probability 1/12. The complete sample space S is:   S = {(B,1), (B,2), (B,3), (B,4), (B,5), (B,6),       (R,1), (R,2), (R,3), (R,4), (R,5), (R,6)} ───────────────────────────── Summary of Answers: a. P(blue ball and rolling a 2) = 1/12 b. P(rolling a 3) = 1/6 c. P(drawing a blue ball) = 1/2 d. P(drawing a red ball or rolling an even number) = 3/4