A student's score depends on being prepared

Question

A student walks into an exam prepared with probability $0.8$ , in which case they score $90$ on average, and unprepared with probability $0.2$ , in which case they score $50$ on average.

What score should you expect?

Accepted Answer

Two branches, prepared or not. The law of total expectation (tower rule) blends the two branch averages by how likely each is: E[	ext{score}] = E\big[\,E[	ext{score} \mid 	ext{preparation}]\,\big]. Step 1, the branch averages are given: prepared 90, unprepared 50. Step 2, weight by the preparation odds, 0.8 prepared and 0.2 unprepared: E[	ext{score}] = 0.8 	imes 90 + 0.2 	imes 50 = 72 + 10 = 82. You should expect a score of 82. Because the student is usually prepared, the blend sits close to the prepared score of 90. Takeaway. Average the outcome in each case, weighted by the probabilit…

A student's score depends on being prepared

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