Question

Two beads are to be independently and randomly selected, one from each of two bags. If \(\frac{2}{7}\) of the beads in one bag and \(\frac{3}{7}\) of the beads in the other bag are yellow, what is the probability that both beads selected will be yellow?

• A. \(\frac{2}{3}\)
• B. \(\frac{5}{7}\)
• C. \(\frac{6}{7}\)
• D. \(\frac{5}{49}\)
• E. \(\frac{6}{49}\)

Hint:

Remember the keyword "and" in probability usually implies multiplication (for independent events), while "or" usually implies addition (for mutually exclusive events).

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem involves calculating the probability of two independent events both occurring. The events are "independent" because the selection of a bead from one bag does not influence the selection from the other bag.
Step 2: Key Formula or Approach:
The probability of two independent events, A and B, both happening is given by the product of their individual probabilities:
\[ P(\text{A and B}) = P(A) \times P(B) \] Step 3: Detailed Explanation:
Let event A be the selection of a yellow bead from the first bag. The probability of this event is given as:
\[ P(A) = \frac{2}{7} \] Let event B be the selection of a yellow bead from the second bag. The probability of this event is given as:
\[ P(B) = \frac{3}{7} \] We want to find the probability that both beads are yellow, which is \(P(\text{A and B})\). Since the events are independent, we multiply their probabilities:
\[ P(\text{both yellow}) = P(A) \times P(B) = \frac{2}{7} \times \frac{3}{7} \] \[ P(\text{both yellow}) = \frac{2 \times 3}{7 \times 7} = \frac{6}{49} \] Step 4: Final Answer:
The probability that both beads selected will be yellow is \(\frac{6}{49}\).