Solve the following problems:
Q: A sports psychologist performed a study of some visualization techniques that she developed to improve athletic performance. She used amateur golfers and amateur tennis players as participants. In her study 70% of the participants were golfers, and the other 30% were tennis players. (No participant was both a golfer and a tennis player.) The visualization techniques seemed quite helpful for both the golfers and the tennis players: 85% of the golfers reported a solid improvement in their performance after using the visualization techniques, and 75% of the tennis players reported a solid improvement in their performance after using the techniques.
Let G denote the event that a randomly chosen participant was a golfer and let the complement of G be the event that a randomly chosen participant was a tennis player. Let I denote the event that a randomly chosen participant reported a solid improvement in performance after using the visualization techniques and let the complement of I be the event that a randomly chosen participant did not report a solid improvement in performance after using the visualization techniques.
Fill in the probabilities to complete the tree diagram below, and then answer the question that follows.
P(I | G) = ___
P(G n I) = ____
P(G) = ____
-
P(G n I) = ____
-
P(I | G) = 0.15
-
P(I | G) = 0.75
-
P(G n I) = ____
P(G) = 0.3
- -
P(G n I) = _____
- -
P(I | G) = ____
Q: What is the probability that a randomly chosen participant reported a solid improvement in performance after using the visualization techniques?