Question

A travel company tracked the preferred destinations of 225 travelers who enjoy different types of leisure activities. The data are displayed in a table.


Beach Mountains City
Outdor Recreation 38 21 12
Reading 32 39 18
TV/Video Games 12 23 30
Part A: Calculate the empirical conditional probability of a traveler who prefers the mountains, given that they enjoy reading. Show all work. (5 points)

Part B: Is a traveler user more or less likely to prefer the mountains if they enjoy reading? Justify the answer mathematically. (5 points)

Answer 1

@toga

Answer 2

Part A: To calculate the empirical conditional probability of a traveler who prefers the mountains given that they enjoy reading, we need to use the formula:

P(Mountains|Reading) = P(Mountains and Reading) / P(Reading)

To calculate the numerator P(Mountains and Reading), we look at the table and see that 39 travelers enjoy reading and prefer the mountains. Therefore,

P(Mountains and Reading) = 39/225

To calculate the denominator P(Reading), we add up the number of travelers who enjoy reading across all three destination types. We get:

P(Reading) = (32+39+18)/225 = 0.42

Therefore, the empirical conditional probability of a traveler who prefers the mountains given that they enjoy reading is:

P(Mountains|Reading) = (39/225) / 0.42 = 0.46

Part B: To determine whether a traveler is more or less likely to prefer the mountains if they enjoy reading, we compare the conditional probability P(Mountains|Reading) to the marginal probability P(Mountains). If P(Mountains|Reading) is greater than P(Mountains), then a traveler who enjoys reading is more likely to prefer the mountains.

To calculate P(Mountains), we add up the number of travelers who prefer the mountains across all three activity types. We get:

P(Mountains) = (21+39+23)/225 = 0.43

Since P(Mountains|Reading) = 0.46 is slighttly greater than P(Mountains) = 0.43, we can conclude that a traveler who enjoys reading is slightly more likely to prefer the mountains.