Consider three jars labelled Jar 1, Jar 2, and Jar 3. Jar 1 contains one red ball, and three green balls. Likewise, Jar 2 contains one red ball, and three green balls. Jar 3 contains two red balls and two green balls. The balls in each jar are mixed thoroughly, and this situation is the starting point for every question below.
a.) Suppose that a ball is drawn at random from Jar 3. What is the probability that this ball is red?
b.) Suppose that a first ball is drawn from Jar 1, and a second abll is drawn from Jar 2. What is the proability that the two balls are green?
c.) Suppose taht one ball is drawn from each jar. What is the probability that exactly one ball out of three is re, and the other two balls are green?
d.) Again, consider the situation where one ball is drawn from each jar. After this drawing, exactly one ball is red and two balls are green. What is the probability that the red ball came from Jar 1.
e.) Consider the following random experiment. First, one jar is selected at random, with the three jars being equally likely. Then, two balls are drawn from this jar without replacement. If the two balls are green, what is the conditional probablity that these were drawn from Jar 1?