Final answer:
To determine if 'the person is female' and 'the person prefers classic rock' are independent, we calculate the probability of each event and their joint probability, then check if the product of the individual probabilities equals the joint probability , and we get that the two events are not independent..
Step-by-step explanation:
To determine if the events "the person is female" and "the person prefers classic rock" are independent, we can use the definition of independence in probability. Two events A and B are independent if and only if P(A and B) = P(A) * P(B). Here, let's let A be the event that the person is female (F), and B the event that the person prefers classic rock (CR).
Firstly we determine P(F), P(CR), and P(F and CR) using the table:
- Total number of females (F) = 17 (Rap) + 37 (Techno) + 36 (Classic Rock) + 15 (Classical) = 105
- Total number of individuals who prefer classic rock (CR) = 42 (Male) + 36 (Female) = 78
- Number of females who prefer classic rock (F and CR) = 36
- Total number of people surveyed = 39 (Male Rap) + 17 (Female Rap) + 17 (Male Techno) + 37 (Female Techno) + 42 (Male Classic Rock) + 36 (Female Classic Rock) + 12 (Male Classical) + 15 (Female Classical) = 215
Now we calculate the probabilities:
- P(F) = Number of females / Total number of people surveyed = 105 / 215
- P(CR) = Number of individuals who prefer classic rock / Total number of people surveyed = 78 / 215
- P(F and CR) = Number of females who prefer classic rock / Total number of people surveyed = 36 / 215
Now we can check if P(F and CR) equals P(F) * P(CR): P(F) * P(CR) = (105 / 215) * (78 / 215) = 8190 / 46225 P(F and CR) = 36 / 215 If P(F and CR) does not equal P(F) * P(CR), then the two events are not independent.