LA1 ANIRUDDH SINGH

ABSTRACT

A binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times. The binomial is a type of distribution that has two possible outcomes (the prefix "bi" means two, or twice). For example, a coin toss has only two possible outcomes: heads or tails and taking a test could have two possible outcomes: pass or fail.

  • The first variable in the binomial formula, n, stands for the number of times the experiment runs.

  • The second variable, p, represents the probability of one specific outcome.

For example, let's suppose you wanted to know the probability of getting a 1 on a die roll. if you were to roll a die 20 times, the probability of rolling a one on any throw is 1/6. Roll twenty times and you have a binomial distribution of (n=20, p=1/6). SUCCESS would be "roll a one" and FAILURE would be "roll anything else." If the outcome in question was the probability of the die landing on an even number, the binomial distribution would then become (n=20, p=1/2). That's because your probability of throwing an even number is one half.

INTRODUCTION

What is a Binomial Distribution? The Bernoulli Distribution.

The binomial distribution is closely related to the Bernoulli distribution. According to Washington State University, "If each Bernoulli trial is independent, then the number of successes in Bernoulli trails has a binomial Distribution. On the other hand, the Bernoulli distribution is the Binomial distribution with n=1."

A Bernoulli distribution is a set of Bernoulli trials. Each Bernoulli trial has one possible outcome, chosen from S, success, or F, failure. In each trial, the probability of success, P(S) = p, is the same. The probability of failure is just 1 minus the probability of success: P(F) = 1 – p. (Remember that "1" is the total probability of an event occurring…probability is always between zero and 1). Finally, all Bernoulli trials are independent from each other and the probability of success doesn't change from trial to trial, even if you have information about the other trials' outcomes.

Real Life Examples

Many instances of binomial distributions can be found in real life. For example, if a new drug is introduced to cure a disease, it either cures the disease (it's successful) or it doesn't cure the disease (it's a failure). If you purchase a lottery ticket, you're either going to win money, or you aren't . Student passing the exam or not . Basically, anything you can think of that can only be a success or a failure can be represented by a binomial distribution.

Formulas

The binomial distribution formula is:

b(x; n, P) = nCx * Px * (1 – P)n – x

Where:
b = binomial probability
x = total number of "successes" (pass or fail, heads or tails etc.)
P = probability of a success on an individual trial
n = number of trials

Note: The binomial distribution formula can also be written in a slightly different way, because nCx = n! / x!(n – x)! (this binomial distribution formula uses factorials (What is a factorial?). "q" in this formula is just the probability of failure (subtract your probability of success from 1).
binomialprobabilityformula

QUESTIONS AND SOLUTIONS

Q. A coin is tossed 10 times. What is the probability of getting exactly 6 heads?

I'm going to use this formula: b(x; n, P) – nCx * Px * (1 – P)n – x
The number of trials (n) is 10
The odds of success ("tossing a heads") is 0.5 (So 1-p = 0.5)
x = 6

P(x=6) = 10C6 * 0.5^6 * 0.5^4 = 210 * 0.015625 * 0.0625 = 0.205078125

Q. 60% of people who purchase sports cars are men.  If 10 sports car owners are randomly selected, find the probabilitythat exactly 7 are men.

Step 1:: Identify 'n' and 'X' from the problem. Using our sample question, n (the number of randomly selected items—in this case, sports car owners are randomly selected) is 10,  and  X (the number you are asked to "find the probability" for) is 7.


Step 2: Figure out the first part of the formula, which is:

n! / (n – X)!  X!

Substituting the variables:

10! / ((10 – 7)! × 7!)

Which equals 120. Set this number aside for a moment.


Step 3: Find "p" the probability of success and "q" the probability of failure. We are given p = 60%, or .6. therefore, the probability of failure is 1 – .6 = .4 (40%).

Step 4: Work the next part of the formula.

pX
= .67
= .0.0279936


Set this number aside while you work the third part of the formula.

Step 5: Work the third part of the formula.

q(.4 – 7)
= .4(10-7)
= .43
= .0.064

Step 6: Multiply the three answers from steps 2, 4 and 5 together.
120  × 0.0279936 × 0.064 = 0.215.

CODING BD IN R

Functions for Binomial Distribution dBinom() Function

This function is used to find probability at a particular value for a data that follows binomial distribution i.e. it finds:

dbinom(3, size = 13, prob = 1 / 6)
[1] 0.2138454

Problem

Suppose there are twelve multiple choice questions in an English class quiz. Each question has five possible answers, and only one of them is correct. Find the probability of having four or less correct answers if a student attempts to answer every question at random.

Solution

Since only one out of five possible answers is correct, the probability of answering a question correctly by random is 1/5=0.2. We can find the probability of having exactly 4 correct answers by random attempts as follows.

dbinom(4, size=12, prob=0.2) 
[1] 0.1328756

To find the probability of having four or less correct answers by random attempts, we apply the function dbinom with x = 0,…,4.

dbinom(0, size=12, prob=0.2) + 
+ dbinom(1, size=12, prob=0.2) + 
+ dbinom(2, size=12, prob=0.2) + 
+ dbinom(3, size=12, prob=0.2) + 
+ dbinom(4, size=12, prob=0.2) 
[1] 0.9274445

Alternatively, we can use the cumulative probability function for binomial distribution pbinom.

```{r}pbinom(4, size=12, prob=0.2)}

```

Answer

The probability of four or less questions answered correctly by random in a twelve question multiple choice quiz is 92.7%.

CONCLUSION

  • Binomial distribution is a discrete probability distribution representing probabilities of a Binomial random variable

  • Binomial random variable represents number of successes in an experiment consisting of a fixed number of independent trials performed in a sequence.

  • An experiment in binomial distribution will consist of a fixed number of independent trials denoted by letter N.

  • A single trial in a binomial experiment is also called as the Bernoulli trial.

  • Binomial probability distribution measures the probability of number of successes that can happen in multiple experiments of N trials.

REFERENCES

Cuemath. "Binomial Distribution Formula Codes:- https://www.r-tutor.com/elementary-statistics/probability-distributions/binomial-distribution Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 531, 1987.
Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 102-103, 1984.
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 108-109, 1992.
Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.
WSU. Retrieved Feb 15, 2016 from: www.stat.washington.edu/peter/341/Hypergeometric%20and%20binomial.pdf