Basic Usage

• dbinom() has three ordered/named arguments
• x, a single integer value or a vector of integer values giving the number(s) of successes.
• size, the number of trials.
• prob, the probability of success on 1 trial

dbinom() returns the probabilty of the input number(s) of successes give the other arguments.

• rbinom() also has three ordered/named arguments
• n,
• size, the number of trials.
• prob, the probability of success on 1 trial

rbinom() returns a vector of length n containing random drawings from the distribution defined by the other arguments.

Examples of dbinom()

Compute the probability of exactly 5 successes out of 10 independent trials where the probability of success on 1 trial is .6.

#Specify arguments in order
dbinom(5,10,.6)

## [1] 0.2006581

# or alternatively, specify names in any order
dbinom(size=10,x=5,prob=.6)

## [1] 0.2006581

Compute the probability of at least 4 and at most 6 successes with 10 trials and probability of success = .6.

# Use three separate calls to dbinom().
dbinom(4,10,.6) + dbinom(5,10,.6) + dbinom(6,10,.6)

## [1] 0.5629575

# Alternatively, use a vector input for x and sum the results using the sum() function.
sum(dbinom(4:6,10,.6))

## [1] 0.5629575

Examples with rbinom()

We can get approximate answers to probability problems using simulated data. The first step is to create a vector of random numbers drawn from the given binomial distribution.

RB = rbinom(100000,size=10,prob=.6)

# Inspect the results
table(RB)

## RB
##     0     1     2     3     4     5     6     7     8     9    10 
##    12   168  1067  4313 11025 20189 25149 21461 12006  4000   610

barplot(table(RB))

We can convert the values in RB into estimated probabilities by dividing the table by 100,000, or more generally by the sum of the counts in the table.

prob.RB = table(RB)/sum(table(RB))

# Look at the probabilities
prob.RB

## RB
##       0       1       2       3       4       5       6       7       8 
## 0.00012 0.00168 0.01067 0.04313 0.11025 0.20189 0.25149 0.21461 0.12006 
##       9      10 
## 0.04000 0.00610

# Check that the sum of the probabilities adds up to 1.0
sum(prob.RB)

## [1] 1

We can use the values in RB directly by remembering that the mean value of a logical expression is the proportion of cases for which the logical expression is true. We can use this to repeat the exercises with above and compare the estimated probabilities with the theoretical values given by dbinom().

mean(RB == 4)

## [1] 0.11025

dbinom(4,10,.6)

## [1] 0.1114767

mean(RB >= 4 & RB <=6)

## [1] 0.56363

sum(dbinom(4:6,10,.6))

## [1] 0.5629575

Exercise 1

Compute the probability of exactly 12 successes out of 17 trials with probability of success = .8.

Exercise 1 Answer

dbinom(12,17,.8)

## [1] 0.1360756

Exercise 2

Compute the probability of more than 5 successes out of 13 trials with probability of success = .2.

Exercise 2 answer

sum(dbinom(6:13,13,.2))

## [1] 0.03003532

Exercise 3

A test consists of 10 true/false questions. To pass the test a student must answer at least 6 questions correctly. If a student guesses on each question, what is the probability that the student will pass the test?

Exercise 3 Answer

sum(dbinom(6:10,10,.5))

## [1] 0.3769531

Exercise 4

A machine has 9 identical components which function independently. The probability that a component will fail is 0.2. The machine will stop working if more than three components fail. Find the probability that the machine will be working.

Exercise 4 Answer

1-sum(dbinom(4:9,9,.2))

## [1] 0.9143583

# or

sum(dbinom(0:3,9,.2))

## [1] 0.9143583

Exercise 5

In a certain college, 33% of the physics majors belong to ethnic minorities. If 10 students are selected at random from the physics majors, what is the probability that no more than 6 belong to an ethnic minority?

Exercise 5 Answer

sum(dbinom(0:6,10,.33))

## [1] 0.9814511

Exercise 6

Find the probability of at least 2 girls in 8 births. Assume that male and female births are equally likely and that the births are independent events.

Exercise 6 Answer

sum(dbinom(2:8,8,.5))

## [1] 0.9648438

Exercise 7

A company purchases shipments of machine components and uses this acceptance sampling plan: Randomly select and test 22 components and accept the whole batch if there are fewer than 3 defectives. If a particular shipment of thousands of components actually has a 7% rate of defects, what is the probability that this whole shipment will be accepted?

Exercise 7 Answer

sum(dbinom(0:2,22,.07))

## [1] 0.8032051

Exercise 8

An airline estimates that 94% of people booked on their flights actually show up. If the airline books 71 people on a flight for which the maximum number is 69, what is the probability that the number of people who show up will exceed the capacity of the plane?

Exercise 8 Answer

sum(dbinom(70:71,71,.94))

## [1] 0.06838378