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.

the probability that both bills are \\$50 bills is
(type an integer or a simplified fraction.)

Explanation:

Identify the sample space structure

Using the Tree Diagrams and Sample Space Construction knowledge points

• First selection options: \(2, 10, 50, 100\)
• Since selection is with replacement, the second selection options for each branch are also: \(2, 10, 50, 100\)
• This yields \(4 \times 4 = 16\) outcomes.
• Diagram A shows 4 main branches, each splitting into 4 sub-branches: \(2, 10, 50, 100\).
• Therefore, Diagram A is the correct tree diagram.

Calculate the probability of selecting two $50 bills

Using the Theoretical Probability knowledge point

• Total outcomes in sample space: \(N = 16\)
• Favorable outcome: \((50, 50)\)
• Number of favorable outcomes: \(n = 1\)
• Probability:

$$ P(\text{both } 50) = \frac{1}{16} $$

Answer:

Question 1

• A. The tree diagram showing 4 initial branches (2, 10, 50, 100), each splitting into 4 identical sub-branches (2, 10, 50, 100). (Correct answer)
• B. The tree diagram showing 4 initial branches, each splitting into only 3 sub-branches (representing selection without replacement).
• C. The tree diagram showing only 2 initial branches.
• D. The tree diagram showing 4 initial branches, each splitting into only 2 sub-branches.

Question 2

The probability that both bills are $50 bills is <blank>\(\frac{1}{16}\)</blank>