L2 C4 S2
S2: Probabilities of SETS of Bernoulli Outcomes
To illustrate how we calculate probabilities for sets of Bernoulli outcomes, we consider a very simple special case. We suppose that N=4 so that there are only 4 Bernoulli Random Variables B1,B2,B3,B4. To make it vivid, consider that you have planted seeds for FOUR mango trees in the four corners of your house. Every seed has probability 30% of germinating and growing up into a tree. Suppose that these corners are names {NSEW} North South East West. A sequence like {1100} represents the outcome that the first two seeds grow into trees while the last two fail. In this case we would have trees in the North and South corners, but nothing in East and West corners.
Every sequence of 4 zeros and ones is a possible outcome of this set of 4 Bernoulli Trials. Since there are 2 choices at each trial, the total number of possible outcomes is 2 x 2 x 2 x 2 = 16. It is useful to put these outcomes into a sequence of Binary Numbers as displayed below:
Question: What is the probability that only one seed germinates, so that we end up with only one tree in our house?
Answer: There are FOUR sequences with only one tree: 0001, 0010, 0100, and 1000. According to the rule we learned in the previous slide, each of these has probability 0.3 x 0.7 x 0.7 x 0.7 = 0.3 x 0.343 =0.1029 =10.29%. These are four different outcomes which are mutally exclusive -- if one of them occurs the others cannot occur. So the probability that any one of the four happens is the SUM of the four probabilities, which is 4 x 10.29%= 41.16% This is the probability of exactly one tree.
We can arrange the 16 possible outcomes in a sequence of binary numbers. The first number her 0000 corresponds to 0, and subsequent numbers are 1,2,3,...,15 all expressed in binary.
EXERCISES: Using the methods explained answer the following questions:
1. What is the probability of getting one tree in the East corner only? {0010}
2. What is the probability of getting one tree in either the East or the West corner? {0010} or {0001}
3. What is the probability of getting exactly one tree in the North South corners and exactly one tree in the East West corners? This is a bit more difficult. NS tree means {10xx} or {01xx} EW tree means {xx10} or {xx01} These two possibilities must be combined and there are 2 x 2 = 4 combinations.
4. What is the probability that EXACTLY two trees grow, in any two corners of the house?
Khan Academy: Introduction to Binary Counting — Important, especially for students unfamiliar with Binary counting