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## Replacing Intuitive Notions With Second-Thought Intutions: Part II

You draw two balls at random from a bag containing blue and red balls.

Suppose you are told that the probability that {one of the balls is blue and the other ball is red} is 1/2.

What can be said about the number of balls of the two colors blue and red in the bag?

A side question: Does it matter if the bag has balls of other colors as well?

Source: Avital & Barbeau, "Intutively Misconceived Solutions to Problems," For the Learning of Mathematics, 1991

Hint: There is not necessarily an equal number of blue and red balls!

Place some blue and red balls in a bag, then start experimenting and recording results (and reflective thoughts).

Solution Commentary: Draw network or graph diagrams to represent pairs in the draw....labeling connecting lines (or arcs) as S (Same) or D (Different).

What happens with the probabilities of a two-ball draw for these situations:

• 2 blue and 1 red
• 2 blue and 2 red
• 3 blue and 2 red
• etc.
A surprising answer should become apparent. Also, can there ever be the same number of blue and red balls in the bag...careful?