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Pete's aunt has collected beads in a variety of different colors and shapes. Pete randomly selects one of...

Pete's aunt has collected beads in a variety of different colors and shapes. Pete randomly selects one of the beads from her collection. The probability that the randomly selected bead is round is 0.4. The probability that the randomly selected bead is round given that it is yellow is 0.7.
Could the event that Pete selects a round bead be independent from the event that he selects a yellow bead? Choose the true statement.
No, the events cannot be independent because 0.4 ≠ 0.7.
No, the events cannot be independent because 0.4 + 0.7 > 1.
Yes, the events must be independent because 0.4 < 0.7.
Yes, the events could be independent because there is not enough information to conclude otherwise.

Answer

Correct choice: No, the events cannot be independent because 0.4 ≠ 0.7.

Independence requires P(round | yellow) = P(round). Since 0.7 ≠ 0.4, knowing that the bead is yellow changes the probability that it is round, so the events are not independent.

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