Mastery Check Solutions

Show What You Know

The school newspaper, The Daily Scoop, recently conducted a survey to find out student participation in extracurricular activities. They surveyed 200 students. Here is what they found:

80 students participate in sports (S) $P (S) = \frac{80}{200} = 0.40$  
60 students participate in music (M) $P (M) = \frac{60}{200} = 0.30$  
30 students participate in both sports and music $P (S and M) = \frac{30}{200} = 0.15$  
The rest of the students don’t participate in either of these activities. $P (none) = \frac{90}{200} = 0.45$

Are the events “participating in sports” and “participating in music” independent? Explain your reasoning.

If events are independent, then $P (A \cap B) = P (A) \cdot P (B)$ .

Participating in sports and in music are not independent because the probabilities $P (A and B) = 0.15$ and $P (A) \cdot P (B) = 0.12$ .

If a student participates in sports, what is the probability that they also participate in music, $P (M | S)$ ?

$P (M | S) = \frac{P (M \cap S)}{P (S)} = \frac{0.15}{0.40} = 0.375$

There is a 37.5% chance that a student participates in music given they play a sport.

What is the probability that a randomly selected student participates in sports AND does not participate in music?

Sports, not both

$0.40 - 0.15 = 0.25$

There is a 25% chance that a student only plays a sport.

Note

Another way to solve:

$80 - 30 = 50 P (S) = \frac{50}{200} = 0.25$

If a student does NOT participate in sports, what is the probability that they participate in music?

$P (S ‘) = 1 - 0.40 = 0.60, P (M \cap S ‘) = P (only M) P (M | S ‘) = \frac{P (M \cap S ‘)}{P (S ‘)} = \frac{0.15}{0.60} = 0.25$

There is a 25% chance that a student participates in music if they are not in sports.

Say What You Know

In your own words, talk about what you have learned using the objectives for this lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Calculate the probability of a dependent event.
Calculate conditional probabilities.
Explain if events are independent or dependent.