chapter 4.3 binomial distribution

binomial experiment

three characteristic of a binomial experiment:
1) There are a fixed number of trials.
• 𝑛 = the number of trials
2) There are only two possible outcomes, called “success” and “failure”, for each trial.
• 𝑝 = the probability of a success on one trial
• 𝑞 = the probability of a failure on one trial
• 𝑝 + 𝑞 = 1
3) The 𝑛 trials are independent and are repeated using identical conditions.

random variable X (binomial distribution)

the number of successes obtained in the 𝑛 independent
trials.
❑ The possible values of 𝑋 are the whole numbers from 0 to 𝑛

binomial probabilities

probability that a
binomial random variable takes any value by adding
probabilities for the different ways of getting exactly that
many successes in 𝑛 observations.

binomial coefficient

number of ways of arranging 𝑘 successes among 𝑛
observations is given by the binomial coefficient
𝑛
𝑘 = 𝑛!
𝑘! 𝑛 − 𝑘 !
for 𝑘 = 0,1,2, ... , 𝑛

This formula uses the factorial notation. For any positive
whole number 𝑛, its factorial 𝑛! Is
𝑛! = 𝑛 × 𝑛 − 1 × 𝑛 − 2 × ... × 3 × 2 × 1
Also, 0! = 1

Read 𝑛
𝑘 as the “binomial coefficient 𝑛
choose 𝑘"

binomial coefficient (𝑛/𝑘)

counts the number of
different ways in which 𝑘 successes can be arranged among 𝑛 observations

binomial probability

If 𝑋 has the binomial distribution with 𝑛 observations
and probability 𝑝 of success on each observation, the
possible values of 𝑋 are 0,1,2 ... , 𝑛.

If 𝑘 is any one of these values,
𝑃 𝑋 = 𝑘 = 𝑛
𝑘 𝑝𝑘 1 − 𝑝 𝑛−𝑘

BINOMIAL PROBABILITY DISTRIBUTION

mean, 𝜇, for a binomial distribution is
𝜇 = 𝑛𝑝

standard deviation, 𝜎, of a binomial distribution is
𝜎 = 𝑛𝑝𝑞

to find q: 1-p

NOTATION FOR THE BINOMIAL

𝑋~𝐵(𝑛, 𝑝)
• Read this as “𝑋 is a random variable with a binomial
distribution”
• The parameters are 𝑛 and 𝑝
o 𝑛 = number of trials
o 𝑝 = probability of a success on each trial

calculator

Go into 2nd DISTR.
To calculate (𝒙 =value): binompdf(𝒏, 𝒑, 𝐧𝐮𝐦𝐛𝐞𝐫)
To calculate 𝑷(𝒙 ≤value): binomcdf(𝒏, 𝒑, 𝐧𝐮𝐦𝐛𝐞𝐫)
Note:
If you want to find 𝑷 𝒙 > 𝐧𝐮𝐦𝐛𝐞𝐫 , use 𝟏 − binomcdf (𝒏, 𝒑, 𝐧𝐮𝐦𝐛𝐞𝐫)

example: trials= 5(n), P=%, x value= total