Question

trevante has a bag of marbles.

• there are 3 yellow marbles (y₁, y₂, y₃) and 1 blue marble (b) in the bag.
• trevante randomly picks a marble out of the bag and then replaces it.
• he repeats this process one more time.

what is a representation of the sample space?
a {y₁, y₂, y₃, b}
b {y₁y₁, y₂y₂, y₃y₃, bb}
c {y₁y₂, y₁y₃, y₁b, y₂y₁, y₂y₃, y₂b, y₃y₁, y₃y₂, y₃b, by₁, by₂, by₃}
d {y₁y₁, y₁y₂, y₁y₃, y₁b, y₂y₁, y₂y₂, y₂y₃, y₂b, y₃y₁, y₃y₂, y₃y₃, y₃b, by₁, by₂, by₃, bb}

Explanation:

Step1: Understand the experiment

Trevante picks a marble, replaces it, and then picks again. So each trial (pick) has 4 possible outcomes: \(Y_1\), \(Y_2\), \(Y_3\), \(B\). Since the first pick and the second pick are independent (because of replacement), we need to consider all possible pairs of outcomes from the two picks.

Step2: List all possible pairs

For the first pick, each of the 4 marbles can be paired with each of the 4 marbles from the second pick.

• If the first pick is \(Y_1\), the second pick can be \(Y_1\), \(Y_2\), \(Y_3\), or \(B\), giving pairs \(Y_1Y_1\), \(Y_1Y_2\), \(Y_1Y_3\), \(Y_1B\).
• If the first pick is \(Y_2\), the second pick can be \(Y_1\), \(Y_2\), \(Y_3\), or \(B\), giving pairs \(Y_2Y_1\), \(Y_2Y_2\), \(Y_2Y_3\), \(Y_2B\).
• If the first pick is \(Y_3\), the second pick can be \(Y_1\), \(Y_2\), \(Y_3\), or \(B\), giving pairs \(Y_3Y_1\), \(Y_3Y_2\), \(Y_3Y_3\), \(Y_3B\).
• If the first pick is \(B\), the second pick can be \(Y_1\), \(Y_2\), \(Y_3\), or \(B\), giving pairs \(BY_1\), \(BY_2\), \(BY_3\), \(BB\).

Combining all these pairs gives the sample space.

Step3: Compare with options

• Option A: This is just the set of marbles, not the sample space for two picks.
• Option B: This only includes pairs where both marbles are the same (e.g., \(Y_1Y_1\)) or \(BB\), missing many other pairs.
• Option C: This is missing pairs where the first and second picks are the same (e.g., \(Y_1Y_1\), \(Y_2Y_2\), \(Y_3Y_3\), \(BB\)).
• Option D: This includes all possible pairs from the two picks, as we listed in Step 2.

Answer:

D. \(\{Y_1Y_1, Y_1Y_2, Y_1Y_3, Y_1B, Y_2Y_1, Y_2Y_2, Y_2Y_3, Y_2B, Y_3Y_1, Y_3Y_2, Y_3Y_3, Y_3B, BY_1, BY_2, BY_3, BB\}\)