Interpret the Mean of a Discrete Random Variable as an Expected Value
-Mamma Temte bakes six pies a day that cost $\$ 2$ each to produce. On $31 \%$ of the days she sells only two pies. On $39 \%$ of the days, she sells 4 pies, and on the remaining $30 \%$ of the days, she sells all six pies. If Mama Temte sells her pies for $\$ 5$ each, what is her expected profit for a day's worth of pies? [Assume that any leftover pies are given away.]
A) $\$ 7.90$
B) $\$ 19.90$
C) $- \$ 7.00$
D) $- \$ 8.02$

Question

Quizplus · Accepted Answer

To calculate the expected profit, first determine the profit for each scenario, then multiply each by its probability and sum the results. The cost to produce 6 pies is $6 \times \$2 = \$12$. For selling 2 pies, the profit is $2 \times \$5 - \$12 = \$10 - \$12 = -\$2$. For selling 4 pies, the profit is $4 \times \$5 - \$12 = \$20 - \$12 = \$8$. For selling 6 pies, the profit is $6 \times \$5 - \$12 = \$30 - \$12 = \$18$. The expected profit is $(-\$2 \times 0.31) + (\$8 \times 0.39) + (\$18 \times 0.30) = -\$0.62 + \$3.12 + \$5.40 = \$7.90$.