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Computational Probability and Statistics

9 Conditional Probability

9.1 Objectives

  1. Define conditional probability and distinguish it from joint probability.
  2. Find a conditional probability using its definition.
  3. Using conditional probability, determine whether two events are independent.
  4. Apply Bayes’ Rule mathematically and via simulation.

9.2 Conditional Probability

So far, we’ve covered the basic axioms of probability, the properties of events (set theory) and counting rules. Another important concept, perhaps one of the most important, is conditional probability. Often, we know a certain event or sequence of events has occurred and we are interested in the probability of another event.

Example:
Suppose you arrive at a rental car counter and they show you a list of available vehicles, and one is picked for you at random. The sample space in this experiment is \[ S=\{\mbox{red sedan}, \mbox{blue sedan}, \mbox{red truck}, \mbox{grey truck}, \mbox{grey SUV}, \mbox{black SUV}, \mbox{blue SUV}\}. \]

What is the probability that a blue vehicle is selected, given a sedan was selected?

Since we know that a sedan was selected, our sample space has been reduced to just “red sedan” and “blue sedan.” The probability of selecting a blue vehicle out of this sample space is simply 1/2.

In set notation, let \(A\) be the event that a blue vehicle is selected. Let \(B\) be the event that a sedan is selected. We are looking for \(\mbox{P}(A \mbox{ given } B)\), which is also written as \(\mbox{P}(A|B)\). By definition, \[ \mbox{P}(A|B)=\frac{\mbox{P}(A \cap B)}{\mbox{P}(B)} \]

It is important to distinguish between the event \(A|B\) and \(A \cap B\). This is a common misunderstanding about probability. \(A \cap B\) is the event that an outcome was selected at random from the total sample space, and that outcome was contained in both \(A\) and \(B\). On the other hand, \(A|B\) assumes the \(B\) has occurred, and an outcome was drawn from the remaining sample space, and that outcome was contained in \(A\).

Another common misunderstanding involves the direction of conditional probability. Specifically, \(A|B\) is NOT the same event as \(B|A\). For example, consider a medical test for a disease. The probability that someone tests positive given they had the disease is different than the probability that someone has the disease given they tested positive. We will explore this example further in our Bayes’ Rule section.

9.3 Independence

Two events, \(A\) and \(B\), are said to be independent if the probability of one occurring does not change whether or not the other has occurred. We looked at this last lesson but now we have another way of looking at it using conditional probabilities. For example, let’s say the probability that a randomly selected student has seen the latest superhero movie is 0.55. What if we randomly select a student and we see that he/she is wearing a black backpack? Does that probability change? Likely not, since movie attendance is probably not related to choice of backpack color. These two events are independent.

Mathematically, \(A\) and \(B\) are considered independent if and only if \[ \mbox{P}(A|B)=\mbox{P}(A) \]

Result: \(A\) and \(B\) are independent if and only if \[ \mbox{P}(A\cap B)=\mbox{P}(A)\mbox{P}(B) \]

This follows from the definition of conditional probability and from above: \[ \mbox{P}(A|B)=\frac{\mbox{P}(A\cap B)}{\mbox{P}(B)}=\mbox{P}(A) \]

Thus, \(\mbox{P}(A\cap B)=\mbox{P}(A)\mbox{P}(B)\).

Example: Consider the example above. Recall events \(A\) and \(B\). Let \(A\) be the event that a blue vehicle is selected. Let \(B\) be the event that a sedan is selected. Are \(A\) and \(B\) independent?

No. First, recall that \(\mbox{P}(A|B)=0.5\). The probability of selecting a blue vehicle (\(\mbox{P}(A)\)) is \(2/7\) (the number of blue vehicles in our sample space divided by 7, the total number vehicles in \(S\)). This value is different from 0.5; thus, \(A\) and \(B\) are not independent.

We could also use the result above to determine whether \(A\) and \(B\) are independent. Note that \(\mbox{P}(A)= 2/7\). Also, we know that \(\mbox{P}(B)=2/7\). So, \(\mbox{P}(A)\mbox{P}(B)=4/49\). But, \(\mbox{P}(A\cap B) = 1/7\), since there is just one blue sedan in the sample space. \(4/49\) is not equal to \(1/7\); thus, \(A\) and \(B\) are not independent.

9.4 Bayes’ Rule

As mentioned in the introduction to this section, \(\mbox{P}(A|B)\) is not the same quantity as \(\mbox{P}(B|A)\). However, if we are given information about \(A|B\) and \(B\), we can use Bayes’ Rule to find \(\mbox{P}(B|A)\). Let \(B_1, B_2, ..., B_n\) be mutually exclusive and exhaustive events and let \(\mbox{P}(A)>0\). Then, \[ \mbox{P}(B_k|A)=\frac{\mbox{P}(A|B_k)\mbox{P}(B_k)}{\sum_{i=1}^n \mbox{P}(A|B_i)\mbox{P}(B_i)} \]

Let’s use an example to dig into where this comes from.

Example:
Suppose a doctor has developed a blood test for a certain rare disease (only one out of every 10,000 people have this disease). After careful and extensive evaluation of this blood test, the doctor determined the test’s sensitivity and specificity.

Sensitivity is the probability of detecting the disease for those who actually have it. Note that this is a conditional probability.

Specificity is the probability of correctly identifying “no disease” for those who do not have it. Again, another conditional probability.

See Figure 9.1 for a visual representation of these terms and others related to what is termed a confusion matrix.

A table of true results and test results for a hypothetical disease. The terminology is included in the table. These ideas are important when evaluating machine learning classification models.

Figure 9.1: A table of true results and test results for a hypothetical disease. The terminology is included in the table. These ideas are important when evaluating machine learning classification models.

In fact, this test had a sensitivity of 100% and a specificity of 99.9%. Now suppose a patient walks in, the doctor administers the blood test, and it returns positive. What is the probability that that patient actually has the disease?

This is a classic example of how probability could be misunderstood. Upon reading this question, you might guess that the answer to our question is quite high. After all, this is a nearly perfect test. After exploring the problem more in depth, we find a different result.

9.4.1 Approach using whole numbers

Without going directly to the formulaic expression above, let’s consider a collection of 100,000 randomly selected people. What do we know?

  1. Based on the prevalence of this disease (one out of every 10,000 people have this disease), we know that 10 of them should have the disease.

  2. This test is perfectly sensitive. Thus, of the 10 people that have the disease, all of them test positive.

  3. This test has a specificity of 99.9%. Of the 99,990 that don’t have the disease, \(0.999*99990\approx 99890\) will test negative. The remaining 100 will test positive.

Thus, of our 100,000 randomly selected people, 110 will test positive. Of these 110, only 10 actually have the disease. Thus, the probability that someone has the disease given they’ve tested positive is actually around \(10/110 = 0.0909\).

9.4.2 Mathematical approach

Now let’s put this in context of Bayes’ Rule as stated above. First, let’s define some events. Let \(D\) be the event that someone has the disease. Thus, \(D'\) would be the event that someone does not have the disease. Similarly, let \(T\) be the event that someone has tested positive. What do we already know? \[ \mbox{P}(D) = 0.0001 \hspace{1cm} \mbox{P}(D')=0.9999 \] \[ \mbox{P}(T|D)= 1 \hspace{1cm} \mbox{P}(T'|D)=0 \] \[ \mbox{P}(T'|D')=0.999 \hspace{1cm} \mbox{P}(T|D') = 0.001 \]

We are looking for \(\mbox{P}(D|T)\), the probability that someone has the disease, given he/she has tested positive. By the definition of conditional probability, \[ \mbox{P}(D|T)=\frac{\mbox{P}(D \cap T)}{\mbox{P}(T)} \]

The numerator can be rewritten, again utilizing the definition of conditional probability: \(\mbox{P}(D\cap T)=\mbox{P}(T|D)\mbox{P}(D)\).

The denominator can be rewritten using the Law of Total Probability (discussed here) and then the definition of conditional probability: \(\mbox{P}(T)=\mbox{P}(T\cap D) + \mbox{P}(T \cap D') = \mbox{P}(T|D)\mbox{P}(D) + \mbox{P}(T|D')\mbox{P}(D')\). So, putting it all together, \[ \mbox{P}(D|T)=\frac{\mbox{P}(T|D)\mbox{P}(D)}{\mbox{P}(T|D)\mbox{P}(D) + \mbox{P}(T|D')\mbox{P}(D')} \]

Now we have stated our problem in the context of quantities we know: \[ \mbox{P}(D|T)=\frac{1\cdot 0.0001}{1\cdot 0.0001 + 0.001\cdot 0.9999} = 0.0909 \]

Note that in the original statement of Bayes’ Rule, we considered \(n\) partitions, \(B_1, B_2,...,B_n\). In this example, we only have two: \(D\) and \(D'\).

9.4.3 Simulation

To do the simulation, we can think of it as flipping a coin. First let’s assume we are pulling 1,000,000 people from the population. The probability that any one person has the disease is 0.0001. We will use rflip() to get the 1,000,000 people and designate as no disease or disease.

set.seed(43)
results <- rflip(1000000,0.0001,summarize = TRUE)
results
##       n heads  tails  prob
## 1 1e+06   100 999900 1e-04

In this case 100 people had the disease. Now let’s find the positive test results. Of the 100 with the disease, all will test positive. Of those without disease, there is a 0.001 probability of testing positive.

rflip(as.numeric(results['tails']),prob=.001,summarize = TRUE)
##        n heads  tails  prob
## 1 999900   959 998941 0.001

Now 959 tested positive. Thus the probability of having the disease given a positive test result is approximately:

100/(100+959)
## [1] 0.09442871

9.5 Homework Problems

  1. Consider: \(A\), \(B\) and \(C\) are events such that \(\mbox{P}(A)=0.5\), \(\mbox{P}(B)=0.3\), \(\mbox{P}(C)=0.4\), \(\mbox{P}(A \cap B)=0.2\), \(\mbox{P}(B \cap C)=0.12\), \(\mbox{P}(A \cap C)=0.1\), and \(\mbox{P}(A \cap B \cap C)=0.05\).
  1. Are \(A\) and \(B\) independent?
  2. Are \(B\) and \(C\) independent?
  1. Suppose I have a biased coin (the probability I flip a heads is 0.6). I flip that coin twice. Assume that the coin is memoryless (flips are independent of one another).
  1. What is the probability that the second flip results in heads?
  2. What is the probability that the second flip results in heads, given the first also resulted in heads?
  3. What is the probability both flips result in heads?
  4. What is the probability exactly one coin flip results in heads?
  5. Now assume I flip the coin five times. What is the probability the result is 5 heads?
  6. What is the probability the result is exactly 2 heads (out of 5 flips)?

  1. Suppose there are three assistants working at a company: Moe, Larry and Curly. All three assist with a filing process. Only one filing assistant is needed at a time. Moe assists 60% of the time, Larry assists 30% of the time and Curly assists the remaining 10% of the time. Occasionally, they make errors (misfiles); Moe has a misfile rate of 0.01, Larry has a misfile rate of 0.025, and Curly has a rate of 0.05. Suppose a misfile was discovered, but it is unknown who was on schedule when it occurred. Who is most likely to have committed the misfile? Calculate the probabilities for each of the three assistants.

  2. You are playing a game where there are two coins. One coin is fair and the other comes up heads 80% of the time. One coin is flipped 3 times and the result is three heads, what is the probability that the coin flipped is the fair coin? You will need to make an assumption about the probability of either coin being selected.

  1. Use Bayes formula to solve this problem.
  2. Use simulation to solve this problem.