Binomial Probability Calculations

STAT 50 - 13 November 2020

123 GO- what’s been on your radio lately?

Outcomes:

• Calculate probabilities for binomial random variables
• Graph the probability mass function PMF for a binomial random variable

Announcements:

• STAT 196K “Analyzing and Processing Big Data” next semester
• Next HW on binomial random variables

For the final section in this class we will study several different well known named random variables. R can calculate standard probabilities for these, as can many other software packages.

The probability mass function

Let X be the sum of n independent bernoulli random variables, each with probability of success p. Then X is distributed as bin(n, p).

The binomial distribution has the following probability mass function:

\[P(X = k) = \binom{n}{k} p^{k}(1-p)^{n-k} \quad \text{for } k = 0, 1, \dots, n\]

Example: Suppose that X ~ bin(10, 0.3). Find P(X = 3).

Implementing the formula:

n = 10
p = 0.3
k = 3
choose(n, k) * p^k * (1-p)^(n-k)

More directly:

dbinom(3, size = 10, prob = 0.3)

123GO: which do you like better?

Plotting

What does this probability mass function look like?

k = 0:10
probability = dbinom(k, size = 10, prob = 0.3)
plot(k, probability, ylim = c(0, 0.4))

p2 = dbinom(k, size = 10, prob = 0.8)
points(k, p2, col = "blue")

Q: What’s the probability that X = 1?

Q: Why is P(X = 10) so small?

Q: Can you give me a value for p for Y ~ bin(10, p) that will make P(Y = 10) > P(X = 10)?

Cumulative Distribution Function

If X ~ bin(n, p) What is the CDF F(k) = P(X <= k)? Just sum up the PMF:

\[P(X \leq i) = \sum_{i = 0}^k \binom{n}{k} p^{k}(1-p)^{n-k}\]

To be concrete, take X ~ bin(10, 0.3) and let’s find P(X <= 3).

We can implement this from the PMF.

d0123 = dbinom(0:3, size = 10, prob = 0.3)
sum(d0123)

Or we can use the CDF directly:

pbinom(3, size = 10, prob = 0.3)