# A random sample of 1200 U.S college students was asked, "What is your perception of your own body? Do you feel that you are overweight, underweight, or about right?"

A random sample of 1200 U.S college students was asked, "What is your perception of your own body? Do you feel that you are overweight, underweight, or about right?" The two-way table below summarizes the data on perceived body image by gender. \begin{array}{l|c|c} & Female & Male \\ \hline About right & 560 & 295 \\ \hline Overweight & 163 & 72 \\ \hline Undenveight & 37 & 73 \end{array}

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joshyoung05M

Definition conditional probability: $$P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{P(A and B)}{P(A)}$$ Solution $$\begin{array}{l|c|c} & Female & Male \\ \hline About right & 560 & 295 & 560 + 295 = 855 \\ \hline Overweight & 163 & 72 & 163 + 72 = 235 \\ \hline Undenveight & 37 & 73 & 37 + 73 = 110 \\ \hline Total & 560 + 163 + 37 + 760 & 295 + 72 + 73 = 440 & 760 + 440 = 1200 \\ \end{array}$$ We note that the table contains information about 1200 U.S. College students (given in the bottom right corner of the table). Moreover, 855 of the 1200 students think their body image is about right, because 855 is mentioned in the row ” About right” and in the column ”Total” of the table. The probability is the number of favorable outcomes divided by the number of possible outcomes: P(About right) $$= \frac{\# of favorable outcomes}{\# of possible outcomes} =$$ $$\frac{855}{1200}$$Next, we note that 560 of the 1200 students are think their body image is about right, because 560 is mentioned in the row ” About right ” and in the column ”Female” of the given table.P(About right and Female) $$= \frac{\# of favorable outcomes}{\# of possible outcomes} =$$  $$\frac{560}{1200}$$ Use the definition of conditional probability: $$P(Female| About\ right) = \frac{P(About\ right\ and\ Female)}{P(About\ right)} = \frac{560/1200}{855/1200} = \frac{560}{855} = \frac{112}{171} \approx 0.6550 = 65.50%$$

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