Question

assume that a hat contains four bills: a \\$2 bill, a \\$10 bill, a \\$50 bill, and a \\$100 bill. each bill is equally likely to be selected. two bills are to be selected at random with replacement. construct a sample space and determine the probability that both bills are \\$50 bills.

which of the figures illustrates the tree diagram for the sample space? choose the correct answer.

Explanation:

Analyze the selection process

Using the Independent Events knowledge point
The experiment consists of selecting two bills at random with replacement from a hat containing four bills: \(\$2\), \(\$10\), \(\$50\), and \(\$100\).
Because selection is with replacement, the set of possible outcomes for both the first and second selection is identical: \(\{2, 10, 50, 100\}\).

Determine the correct tree diagram

Using the Independent Events knowledge point

• The first stage of the tree diagram must branch into four outcomes: \(2\), \(10\), \(50\), and \(100\).
• Since selection is with replacement, each of these first-stage branches must branch into the same four outcomes: \(2\), \(10\), \(50\), and \(100\).
• Option A correctly shows four primary branches (\(2\), \(10\), \(50\), \(100\)), and each of these branches has four secondary branches (\(2\), \(10\), \(50\), \(100\)).

Calculate the probability

Using the Multiplication Rule for Independent Events knowledge point

$$ P(\text{both } \$50) = P(\text{First is } \$50) \times P(\text{Second is } \$50) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} $$

Answer:

• (A) First stage branches to 2, 10, 50, 100; each second stage branches to 2, 10, 50, 100 (Correct answer)
• (B) First stage branches to 2, 10, 50, 100; second stage branches omit the selected bill (without replacement)
• (C) Incomplete tree diagram showing only 2 and 10 in the first stage
• (D) First stage branches to 2, 10, 50, 100; second stage branches show only two outcomes per branch