3 In a random sample of male and female graduates of the New

3. In a random sample of male and female graduates of the New York School for the Arts between the ages of 22-35 you know that:
   the probability a man is a ballet dancer is 0.245.
   the probability a woman is a musical performer is 0.365
   the probability a man is a musical performer, given that he’s a ballet dancer is 0.250.
   the probability a woman is an actress is 0.550.
   the probability a woman is an actress, given that she’s a musical performer is .785.
Compute the following probabilities: (6 points)
a. P (man is a ballet dancer and a musical performer)
b. P (woman is an actress and a musical performer)
c. P (woman is an actress or a musical performer)

4. Two hundred single sport athletes were cross-classified according to gender as follows:
   Swimmer   Fencer   Boxer
Male   25   60   25
Female   20   50   20

An athlete is selected at random. (12 points)

a.   What is the probability that the athlete is a fencer?
b.   What is the probability that the athlete is a female fencer?
c.   If the athlete is known to be a fencer, what is the probability that the athlete is female?
d.   What is the probability that the athlete is male or a swimmer or both?
e.   Are the events being a female and being a swimmer mutually exclusive? Justify your answer.
f.   Are the events being a female and being a fencer independent? Justify your answer.

Solution

3)a) 0.250

b)0.785

c)0.550+0.356 = 0.906

4)a)110/200

b)50/200

c)50/110

d)130/200

e) no because a female can also be a swimmer.