Solution to problem 1(b) from exam ST2133 ZA, 2019
Simple tools for combinatorial problems
Before solving the problem, it is useful to compare the case of independent events with that of dependent events.
Suppose the events are independent (in the context of the problem, it will be drawings with replacement). Then by definition the joint probability is the product of individual probabilities:
Now assume that the event occurs first, occurs second, ...., occurs last and each subsequent event depends on the previous one (as in the case of drawings without replacement). Then
Similarly, by multiplying and dividing many times, we get
Equation (2) is called a chain rule for probability. Several of my students have been able to solve the problem without explicitly using (2). It is advisable to use (2) or other relevant theoretical properties to achieve clarity and avoid errors.
Problem statement and solution
Suppose there are red balls and green balls in a bag. All balls with the same color are indistinguishable.
Suppose one ball is drawn at a time at random with replacement from the bag. Let be the number of balls drawn until a red ball is obtained (including the red ball). Write down the probability mass function of .
Solution. Most students answer that this is a hypergeometric distribution with probabilities given by where is the probability of success. Without specifying (the probability of drawing a red ball) the answer is incomplete. Since we have
Now suppose one ball is drawn at random at a time without replacement from the bag. Let be the number of balls drawn until a red ball is obtained (including the red ball). Write down the probability mass function of .
Solution. Let us denote the event that the th ball is red and the event that the th ball is green, respectively. Note that the only way appears is by obtaining before it. Hence, equals Besides, are the only (mutually exclusive) possibilities and it remains to find their probabilities.
Next, using (2)
The results can be summarized in a table:
This distribution is not of one of standard types.
Suppose two balls are drawn at the same time at random with replacement from the bag. Let denote the number of these double draws performed until two green balls are obtained. Show that the probability of drawing two green balls is
Hence, show that the probability mass function for is
Solution. Using the same notation as before,
All other events (two red balls or one green and one red) are considered a failure. Thus we have a hypergeometric distribution with