## I do not understand binomial distribution and i need help?

Posted under Female CEO by Marcus Pottea on August 31st, 2013 5:24 pm

I don’t know how to do any of this at all…how do i do it?

Suppose, based on some information source, it is found that approximately about 10% of female employees with MBA degree and a job experience of at least 10 years have become the CEO’s of major corporations. Consider a group of 20 female employees in U.S. major corporations with MBA degree and at least 10 years of job experience, answer the following questions.

i) Explain how this situation fits the binomial distribution through the three basic characteristics of the binomial distribution.
ii) What are the parameters of this binomial distribution?
iii) Find the probability that exactly two will become CEO’s.
iv) Find the probability that at least two will become CEO’s.
v) Compute and interpret the mean and standard deviation of the distribution using formulas.

By definition:" the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p."

Let’s assume your experiments are independent (meaning here the chance of one female employee becoming a CEO has no impact on the others’ chances. Not entirely true but whatever)
the yes/no experiment is "does she become a CEO". Either she does or she does not. pretty simple.
You have 20 independent experiments (we follow the careers of 20 females and assume their chances of becoming CEO are independent of each other)
So each independent experiment can be considered a Bernoulli trial with a fixed probability, which fits the description of a binomial distribution.
Your parameters are n=20 (number of trials) and p= 0.1 (10% chance of success on each trial)

Probability that k become CEO: P(X=k) = n!/(k!(n-k)!)p^k(1-p)^(n-k)
Application for k=2: P(X=2) = 20!/(18!2!)*0.1^2*0.9^18 = 0.285 or 28.5%

Probability that at least 2 become CEO = 1 – P(X=0) – P(X=1) (the reverse probability that either none or just one become CEO)
P(X>1) = 1 – 0.1^20 – 20!/(19!1!)0.1*0.9^19 = 0.608 = 60.8%

Formulas for mean and standard deviation in a binomial distribution:
mean = np = 20*0.1 = 2
variance = np(1-p) = 20*0.1*0.9 = 1.8
standard deviation = square root of variance = 1.34
Interpretation: you should see on average 2 female employees in your sample of 20 become CEOs.
You are about 95% sure there will be between 0 and 5 female employees become CEO (mean +/- 2 standard deviations)

### One Response to “I do not understand binomial distribution and i need help?”

1. By definition:" the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p."

Let’s assume your experiments are independent (meaning here the chance of one female employee becoming a CEO has no impact on the others’ chances. Not entirely true but whatever)
the yes/no experiment is "does she become a CEO". Either she does or she does not. pretty simple.
You have 20 independent experiments (we follow the careers of 20 females and assume their chances of becoming CEO are independent of each other)
So each independent experiment can be considered a Bernoulli trial with a fixed probability, which fits the description of a binomial distribution.
Your parameters are n=20 (number of trials) and p= 0.1 (10% chance of success on each trial)

Probability that k become CEO: P(X=k) = n!/(k!(n-k)!)p^k(1-p)^(n-k)
Application for k=2: P(X=2) = 20!/(18!2!)*0.1^2*0.9^18 = 0.285 or 28.5%

Probability that at least 2 become CEO = 1 – P(X=0) – P(X=1) (the reverse probability that either none or just one become CEO)
P(X>1) = 1 – 0.1^20 – 20!/(19!1!)0.1*0.9^19 = 0.608 = 60.8%

Formulas for mean and standard deviation in a binomial distribution:
mean = np = 20*0.1 = 2
variance = np(1-p) = 20*0.1*0.9 = 1.8
standard deviation = square root of variance = 1.34
Interpretation: you should see on average 2 female employees in your sample of 20 become CEOs.
You are about 95% sure there will be between 0 and 5 female employees become CEO (mean +/- 2 standard deviations)
References :
http://en.wikipedia.org/wiki/Binomial_distribution