A big or small bag of marbles

Question

A crate holds big bags and small bags of marbles. Any bag you grab is a big bag with probability $\tfrac14$ (averaging $8$ marbles) or a small bag with probability $\tfrac34$ (averaging $3$ marbles).

How many marbles should you expect in the bag you grab?

Accepted Answer

Two branches, big bag or small bag. The law of total expectation (tower rule) blends their averages by how likely each bag is: E[	ext{marbles}] = E\big[\,E[	ext{marbles} \mid 	ext{bag}]\,\big]. Step 1, the branch averages are given: big bag 8, small bag 3. Step 2, weight by the bag odds, 	frac14 big and 	frac34 small: E[	ext{marbles}] = 	frac14 	imes 8 + 	frac34 	imes 3 = 2 + 2.25 = 4.25. You should expect 4.25 marbles, that is \frac{17}{4}. Small bags are three times as common, so the blend leans toward the small-bag average. Takeaway. Weight each bag type's average by its probabil…

A big or small bag of marbles

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