Probability Distributions
In this section we will practice calculating probabilities using different probability distributions. We will start with the binomial distribution then move on to the normal distribution.
Binomial Distribution
We will begin with the binomial distribution.
We are interested in studying hand hygiene practices at a local hospital. Suppose that the probability of a healthcare worker following proper hand hygiene practices is 75%. We plan to observe 15 healthcare workers and will record if they comply with health hygiene recommendations or not.
We are going to start by empirically drawing from a binomial distribution to get a sense of what it looks like. Then we will use R functions to calculate probabilities directly from a binomial distribution.
Fill in the code below to randomly draw one sample (n = 1) from a binomial distribution with 15 healthcare workers and a p = 0.75 probability of compliance with hand hygiene recommendations.
# sample one time from a binomial distribution (note, n = 1 indicates we are observing one set of 15 healthcare workers)
rbinom(n = 1, size = , p = )
# sample one time from a binomial distribution (note, n = 1 indicates we are observing one set of 15 healthcare workers)
rbinom(n = 1, size = 15, p = 0.75)If you completed the blanks above correctly, should see a number between 0 and 15. This number corresponds to the number out of 15 healthcare workers followed hand hygiene recommendations.
Since this is a random process, we will all see different numbers.
In fact, try clicking the “Run Code” button above a few times. You should see different numbers appear when you run the code multiple times. This is what the code is supposed to do, since it is randomly generating values from this distribution.
For instructional purposes, it will be easier if we all get the same answers. We will show a strategy to address this next.
Setting Random Seeds
To help us all get the same answers, we can add a line of code to do something called “setting the random seed.” This is a bit technical, but what you should know is that if we all set the seed to the same number, we should get the same answers so our work will be more reproducible (don’t worry, the results are still random–they’re just a bit easier to keep track of!).
# set random seed for reproducibility (I'm setting it to 123, but the number doesn't matter!)
set.seed(123)
# sample one time from a binomial distribution (note, n = 1 indicates we are observing one set of 15 healthcare workers)
rbinom(n = 1, size = 15, p = 0.75)
Now we should all get the number 12.
Drawing Multiple Samples
So far, we have only drawn from this binomial distribution a few times. To get a sense of the shape of the distribution, we will need to sample from it many times. To do this, we will change the argument n in the code above.
Fill in the code below to draw from a binomial distribution multiple times. Start by drawing 5 samples from the distribution, then 10, then 30. Hit “Run Code” each time to see how changing the argument n impacts the output.
# set random seed for reproducibility (I'm setting it to 123, but the number doesn't matter!)
set.seed(123)
# sample from a binomial distribution
rbinom(n = , size = 15, p = 0.75)
# set random seed for reproducibility (I'm setting it to 123, but the number doesn't matter!)
set.seed(123)
# sample from a binomial distribution (5 times)
rbinom(n = 5, size = 15, p = 0.75)[1] 12 10 12 9 9
# sample from a binomial distribution (10 times)
rbinom(n = 10, size = 15, p = 0.75)[1] 12 10 12 9 9 14 11 9 11 12
# sample from a binomial distribution (30 times)
rbinom(n = 30, size = 15, p = 0.75)[1] 12 10 12 9 9 14 11 9 11 12 8 12 11 11 13 9 12 14 12 8 9 10 11 7 11
[26] 10 11 11 12 13
Visualizing a Binomial Distribution
Now, let’s draw from the distribution 10,000 times (imagine observing 10,000 different sets of 15 healthcare workers), save the results in a hand hygiene data frame called hh , and visualize the distribution.
Fill in the code below to draw 10,000 samples from the same binomial distribution. The outcomes of these 10,000 draws will be saved in a data frame called hh, which can then be used to plot the distribution.
# set random seed for reproducibility (I'm setting it to 123, but the number doesn't matter!)
set.seed(123)
# sample from a binomial distribution 10,000 times
hh = data.frame(compliant = rbinom(n = , size = , p = ))
# plot the distribution
hh %>%
ggplot(aes(x = )) +
geom_bar(fill = "blue") +
theme_bw()We can also make a table of the number of times each outcome was observed
# table of outcome frequencies
xtabs(~ compliant, data = hh)Describe the distribution. Which values are the most common? What is the highest value observed? What is the lowest value observed?
Empirical Probabilities from a Binomial Distribution
Now let’s calculate some (empirical) probabilities using our simulated distribution. These won’t be the true theoretical probabilities, but they should be close!
- How often do we see all 15 healthcare workers follow hand hygiene recommendations?
# probability all 15
sum(hh$compliant == ) / 10000# probability all 15
sum(hh$compliant == 15) / 10000[1] 0.013
- How often do fewer than 10 healthcare workers follow hand hygiene recommendations?
# probability fewer than 10
sum(hh$compliant < ) / 10000
# probability fewer than 10
sum(hh$compliant < 10) / 10000[1] 0.1436
- How often do at least 12 healthcare workers follow hand hygiene recommendations?
# probability at least 12
sum(hh$compliant ) / 10000# probability at least 12
sum(hh$compliant >= 12) / 10000
# alternatively:
# probability at least 12
sum(hh$compliant > 11) / 10000[1] 0.4641
- How often do exactly 2 healthcare workers follow hand hygiene recommendations?
# probability exactly 2
sum(hh$compliant ) / 10000# probability exactly 2
sum(hh$compliant == 2) / 10000
sum(hh$compliant > 11) / 10000[1] 0
Across 10,000 tries, we never observe only 2 out of 15!
Theoretical Probabilities
We saw that only 1.3% of the simulated distribution has a value of 15, 14.36% had a value under 10, and 46.41% had a value of at least 12. Let’s see what these probabilities should be based on the true, theoretical binomial distribution.
- What is the true probability that all 15 healthcare workers follow hand hygiene recommendations?
# probability all 15
dbinom(x = , size = 15, prob = 0.75)# probability all 15
dbinom(x = 15, size = 15, prob = 0.75)[1] 0.01336346
- What is the true probability that fewer than 10 healthcare workers follow hand hygiene recommendations?
# probability fewer than 10
pbinom(q = , size = 15, prob = 0.75)# probability fewer than 10
pbinom(q = 9, size = 15, prob = 0.75)[1] 0.1483681
Recall that pbinom(q) gives P(X≤q), so if we want strictly less than 10 this is the same as less than or equal to 9
- What is the true probability that at least 12 healthcare workers follow hand hygiene recommendations?
# probability at least 12
pbinom(q = , size = 15, prob = 0.75, lower.tail = )# probability at least 12
pbinom(q = 11, size = 15, prob = 0.75, lower.tail = F)
# alternatively:
1 - pbinom(q = 11, size = 15, prob = 0.75)[1] 0.4612869
lower.tail = F means that we are calculating the upper tail; that is, P(X>q) or 1-P(X≤q). Again, since we want to include 12, at least 12 means strictly larger than 11.
- What is the true probability that exactly 2 healthcare workers follow hand hygiene recommendations?
# probability exactly 2
dbinom(x = , size = 15, prob = 0.75)# probability exactly 2
dbinom(x = 2, size = 15, prob = 0.75)[1] 8.800998e-07
It is very unlikely to observe 2 out of 15 for this binomial distribution since p=0.75 is quite high.
Notice that the probabilities we have calculated from the theoretical binomial distribution are very similar to the empirical probabilities we calculated from our simulated distribution. This is by design!
We simulated a large sample from a specific distribution to help us visualize what the distribution looks like; that is, to help us understand the relative frequencies of different outcome values from this particular distribution.