Final answer:
To determine the probability that a student likes either jazz or country but not rock, you subtract those who like all three and those who like rock and another type from the individual totals, then add the remaining numbers for jazz and country. The probability is 220 out of 500, which is 44%.
Step-by-step explanation:
To find the probability that a randomly selected student likes jazz or country but not rock, we need to calculate the number of students who like either jazz or country but exclude those who also like rock. We'll use the given numbers for the total students surveyed, and those who like each type and their intersections, to fill out a Venn diagram and then perform the calculations.
First, we put the 11 who like all three types of music in the intersection of all three circles. Next, we add those who like two types: since 30 like rock and country, after accounting for the 11 that like all three, 30 - 11 = 19 like just rock and country. We do the same for rock and jazz (25 - 11 = 14) and country and jazz (31 - 11 = 20). Now we can find the number of students who like just one type: for rock (204 - 19 - 14 - 11 = 160), for country (170 - 19 - 20 - 11 = 120), and for jazz (125 - 14 - 20 - 11 = 80).
Finally, we want students who like jazz or country but not rock, so we add those who like only jazz or only country plus those who like both jazz and country but not rock: 80 (jazz only) + 120 (country only) + 20 (country and jazz only) = 220.
The probability is then 220 out of the total 500 students surveyed: P(Jazz or Country but not Rock) = 220 / 500 = 0.44 or 44%.