Binomial Distribution

Abdullah Al Mahmud

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Concept

  • Success/failure
  • Detection of a disease is a success!

Example

The probability of choosing statistics is r round(8/50,2). If 20 students are randomly selected, what is the probability that

  • at least 5 would choose the subject?
  • exactly 2 would choose the subject?
  • none will choose?

Function

\(P(x) = ^nC_x \space p^x(1-p)^{n-x}\)

\(1-p=q\)

  • \(p \to\) probability of success
  • \(q \to\) probability of failure
  • \(n \to\) number of Bernoulli trials
  • \(x \to\) no. of success

Bernoulli Trial

  • Introduced by James Bernoulli
  • Independent trials with dichotomous outcomes (success/failure)
  • Coin toss, disease detection

Bernoulli Distribution

\(p^xq^{1-x}\)

If n = 1 in binomial distribution

Assumptions of Binomial Distribution

  • Trials are independent
  • p, q remain constant for each trial
  • Outcomes are dichotomous
  • X counts number of success

Properties of Binomial Distribution

  • Mean: \(np\)
  • Variance: \(npq\)
  • Skewness, \(\beta_1 = \frac{(q-p)^2}{npq}\)
  • Kurtosis, \(\beta_2 = 3 + \frac{1-6pq}{npq}\)

Derivation of pmf

\(P(SF SF \cdots) = P(S) \times P(F) \cdots\)

  • \((p \times p \cdots \times p)(\times q \times q \cdots \times q)\)
  • \(p^xq^{n-x}\)
  • \(^nC_x \space p^xq^{n-x}\) (any x combination from n)
  • 2 items from a, b, c \(\to ab, ac, bc \to \space ^3C_2\)

Derivation of Mean

\(\displaystyle E(X) = \sum_{x=0}^n x\cdot p(x)\) (think: why zero?)

  • \(0 \cdot p(0) + \cdots\)
  • \(np\{q^{n-1}+n(n-1)pq^{n-2}+\cdots+p^{n-1}\}\)
  • \(np(q+p)^{n-1} [binomial \ theorem]\)
  • \(np\) [p+q=1]

Uses of Binomial Distribution

  • Quality control
  • Test of fit

Binomial to Poisson

See here

Binomial to Noraml

If \(n \to \infty\) and \(np\ge 5, nq \ge 5\)

Problem 01

The mean and sd of a binomial distribution are 40 and 6. Find

  1. \(n,p,q\)
  2. \(P(x)\)
  3. \(P(x=0), P(x\ge2), P(x\lt3)\)
  • \(np = 40\)
  • \(\sqrt{npq} = 6\)
  • \(np(1-p)=36 \to 40(1-p)=36 \to 1-p=.9\)

Problem 02

An unbiased coin is tossed 10 times. Find the probability of

  1. 3 heads
  2. at least 2 heads
  3. at most 1 head

Problem 03

The probability that it rains on a particular day is 0.2. If 10 days are observed, what is the probability that it would rain on

  1. all 10 days
  2. no days
  3. at least 2 days
  4. on exactly four days

Problem 04

In El Clasico, the probability that Barcelona wins is 0.39 and that Real Madrid wins is 0.42. In the next 5 matches, what is the probability that

  1. Barcelona wins all the matches?
  2. Real wins all the matches
  3. Barcelona wins at least 2 matches
  4. 2 matches are drawn
  5. more than 3 matches are drawn
  • \(p_1=\)Barca wins, \(p_2 =\) Real wins, \(p_3 =\)Match drawn

Box/Packet

A company produces smartphones, and it is known that 5% of the smartphones have a manufacturing defect on average. The company ships 15 smartphones in each box, and a retailer purchases 500 boxes.

  1. What is the probability that the number of defective smartphones in a box is at least 1?
  2. How many boxes are expected to contain exactly 2 defective smartphones?
  • n = 15 and N = 500

Department

A university finds that 8% of its printed course materials contain typographical errors. Each department orders 50 materials per term. Across the university, 600 departments place orders.

  1. What is the probability that a randomly selected department receives at most 3 erroneous materials?
  2. How many departments are expected to receive at least 5 erroneous materials?

Question Bank

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