Grab an apple or a banana from the basket

Question

A basket is 70\% apples and 30\% bananas. You grab one piece of fruit without looking. An apple averages 120 calories, and a banana averages 100 calories. How many calories should you expect the fruit you grabbed to have?

Accepted Answer

Two stages: which fruit you happen to grab, then that fruit's calorie count. Condition on the fruit type and apply the tower rule: E[	ext{calories}] = E\big[\,E[	ext{calories} \mid 	ext{fruit}]\,\big]. Step 1, the conditional averages are given: apple 120, banana 100. Step 2, blend by how likely each fruit is to be grabbed: E[	ext{calories}] = 0.70 	imes 120 + 0.30 	imes 100 = 84 + 30 = 114. You expect 114 calories. Because apples are both more common and higher in calories, the blend sits close to the apple value. Takeaway. A mixture's mean is the selection-weighted average of the grou…

Grab an apple or a banana from the basket

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