Brent claims that more students at his school prefer hot dogs over hamburgers. His friend Shaun doesn’t agree. To settle their argument they conduct a random survey of 542 males and 635 females. When the survey results come in, hamburgers are chosen by 34.2% of the females and hot dogs are preferred by 184 males.

Brent, satisfied with the results, proclaims “Yes, hot dogs it is!” Shaun is dismayed by the results and believes that the results are influenced by surveying too many females. How can Shaun use probabilities to convince Brent that the outcome is influenced by gender?

To prove that the results are influenced by gender you can calculate both the probability of preferring hot dogs and the conditional probability of preferring a hot dog given that is a female.

If the two results are different the probability of preferring hot dog is dependent on whether the person is a female or a male.

The probability of preferring hot dogs given that is a female is stated by the problem: 34.2%.

The probability of preferring hot dogs by the whole sample is:

Number of males that prefer hot dogs: 184 (stated by the problem)
Number of females that prefer hot dogs:

100% - 34.2% = 65.8%

65.8% of 635 = 0.658 × 635 = 417.83 ≈ 418

Samples size: 542 males + 635 females = 1177

Probability of preferring hot dogs =

number of students that preffer hot dogs / number of students =

(184 + 418) / 1177 = 602 / 1177 = 0.5115 ≈ 51.2%

Thus, the probability of preferring hot dogs given that the student is a female (34.2%) is different from the probability of preferring hot dog for the whole sample, making the results dependent of the gender.