Jackson B. answered • 07/01/20
Probability Tutor
a) The first thing to do in this question is to find the range of possible values that can result from the experiment. There are two test subjects and both of them can either have attached earlobes or not have attached earlobes. There are three possibilities: zero subjects have attached earlobes, one out of the two has attached earlobes, or both have attached earlobes. Therefore, the range is [0,2] or 0, 1, and 2.
The easiest way to picture the answer for this problem is through a probability tree. We test the first subject, and two paths emerge: the subject has attached earlobes (37% likely) or the subject does not have attached earlobes (63% likely). By the first path, we can write 0.37, and by the second path, we can write 0.63. Each of these branches now has two other possibilities as the second subject must be tested. Just like we did before, two paths emerge from each path, the first with 0.37 written by it and the other with 0.63 written by it.
b) To find the probability of each path happening, we must multiply the two probabilities written on the path. For instance, the probability of the top path happening (both subjects have attached earlobes) is (0.37)*(0.37) = 0.1369. There are two paths which represent only one subject having attached earlobes, and the probability of each of them happening is (0.37)*(0.63) = 0.2331. As they both reach the same outcome, we can add them to find the probability that only one subject has attached earlobes, which is 0.4662. Finally, the probability that neither one has attached earlobes is (0.63)*(0.63) = 0.3969.
Here is the data vaguely arranged into the best probability distribution I could come up with:
X: | 0 | 1 | 2
--------|------------|--------------|----------------
P(X): | 0.3969 | 0.4662 | 0.1369
To check our answer, 0.3969 + 0.4662 + 0.1369 = 1.0000
There is another, and perhaps faster, way to answer this problem without a probability tree. What if the problem selected 10 people at random or even 100 people? Even with only 10 people, our probability tree would consist of 20 individual branches each representing different possibilities! Instead of drawing all that out and analyzing it, there is a formula which makes this problem easier for larger sample sizes.
If n is the number of events (2 in the context of the problem as two people were randomly selected), r is the number whose probability we want to find (0, 1, or 2 in the context of the problem), and p is the probability of the event occurring (0.37), we can find the probability of r people having attached earlobes through the following formula: P(r) = nCr * (p)r * (1-p)n-r For instance, P(2) = 2C2 * 0.372 * 0.630 = 1 * 0.37 * 0.37 * 1 = 0.1369, which is the same answer we found earlier for P(2).
If you would like to see a much more complicated problem that utilizes this technique, click the link below to see a video from my YouTube channel:
