# Binomial Distribution What the binomial distribution is What the binomial distribution is How to recognise situations where the binomial distribution applies.

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Binomial Distribution What the binomial distribution is What the binomial distribution is How to recognise situations where the binomial distribution applies How to recognise situations where the binomial distribution applies How to find probabilities for a given binomial distribution, by calculation and from tables How to find probabilities for a given binomial distribution, by calculation and from tables

When to use the binomial distribution Independent variables Independent variables

Pascals Triangle (a+b) n 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 10 ways to get to the 3 rd position numbering each of the terms from 0 to 5. this can also be calculated by using nCr button on your calculator 5 C 2 =10 nCr 5C05C05C05C01 5C15C15C15C15 5C25C25C25C210 5C35C35C35C310 5C45C45C45C45 5C55C55C55C51

Pascals Triangle (a+b) n 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 nCr n!÷(c!x(n-c)!) 5C05C05C05C0 5!÷(0!x5!) 1 5C15C15C15C1 5!÷(1!x4!) 5 5C25C25C25C2 5!÷(2!x3!) 10 5C35C35C35C3 5!÷(3!x2!) 10 5C45C45C45C4 5!÷(4!x1!) 5 5C55C55C55C5 5!÷(5!x0!) 1

A coin is tossed 7 times. Find the probability of getting exactly 3 heads. We could do Pascal's triangle or we could calculate: 7 C 3 x (P(H)) 7 7 C 3 x (P(H)) 7 The probability of getting a head is ½

TASK Exercise A Page 61 Exercise A Page 61

Unequal Probabilities A dice is rolled 5 times A dice is rolled 5 times What is the probability it will show 6 exactly 3 times? What is the probability it will show 6 exactly 3 times?P(6)=5/6P(6)=1/6

Task / Homework Exercise B Page 62 Exercise B Page 62

The Binomial distribution is all about success and failure. When to use the Binomial Distribution –A fixed number of trials –Only two outcomes –(true, false; heads tails; girl,boy; six, not six …..) –Each trial is independent IF the random variable X has Binomial distribution, then we write X ̴ B(n,p)

Sometimes you have to use the Binomial Formula

Eggs are packed in boxes of 12. The probability that each egg is broken is 0.35 Find the probability in a random box of eggs: there are 4 broken eggs

Task / homework Exercise C Page 65 Exercise C Page 65

Eggs are packed in boxes of 12. The probability that each egg is broken is 0.35 Find the probability in a random box of eggs: There are less than 3 broken eggs

USING TABLES of the Binomial distribution An easier way to add up binomial probabilities is to use the cumulative binomial tables Find the probability of getting 3 successes in 6 trials, when n=6 and p=0.3 n=6x0123456 P=0.3P(X=x)0.11760.42020.74430.92950.98910.99931.000

n=6x0123456 P=0.3P(X=x)0.11760.42020.74430.92950.98910.99931.000 http://assets.cambridge.org/97805216/05397/excerpt/9780521605397_excerpt.pdf The probability of getting 3 or fewer successes is found by adding: P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.1176 + 0.3026 + 0.3241 + 0.1852 = 0.9295 The probability of getting 3 or fewer successes is found by adding: P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.1176 + 0.3026 + 0.3241 + 0.1852 = 0.9295 This is a cumulative probability.

Task / homework Exercise D page 67 Exercise D page 67

Mean variance and standard deviation μ = Σ x x P(X= x )=mean μ = Σ x x P(X= x )=mean This is the description of how to get the mean of a discrete and random variable defined in previous chapter. This is the description of how to get the mean of a discrete and random variable defined in previous chapter. The mean of a random variable whos distribution is B(n,p) is given as: The mean of a random variable whos distribution is B(n,p) is given as: μ =np μ =np

Mean, variance & standard deviation σ²=Σ x ² x P(X= x ) - μ² σ²=Σ x ² x P(X= x ) - μ² is the definition of variance, from the last chapter of a discrete random variable. is the definition of variance, from the last chapter of a discrete random variable. The variance of a random variable whose distribution is B(n,p) The variance of a random variable whose distribution is B(n,p) σ²= np(1-p) σ²= np(1-p) σ= σ=

TASK / HOMEWORK Exercise E Exercise E Mixed Questions Mixed Questions Test Your self Test Your self

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