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Binomial Distribution

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

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Binomial Distribution

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

  2. When to use the binomial distribution • Independent variables

  3. Pascal’s 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 3rd position numbering each of the terms from 0 to 5. this can also be calculated by using nCr button on your calculator 5C2=10

  4. Pascal’s Triangle (a+b)n 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

  5. A coin is tossed 7 times. Find the probability of getting exactly 3 heads. We could do Pascal's triangle or we could calculate: 7C3 x (P(H))7 The probability of getting a head is ½

  6. TASK • Exercise A Page 61

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

  8. Task / Homework • Exercise B Page 62

  9. 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)

  10. Sometimes you have to use the Binomial Formula

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

  12. Task / homework • Exercise C Page 65

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

  14. 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

  15. http://assets.cambridge.org/97805216/05397/excerpt/9780521605397_excerpt.pdfhttp://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.

  16. Task / homework • Exercise D page 67

  17. Mean variance and standard deviation • μ = Σxx P(X=x)=mean • 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: • μ =np

  18. Mean, variance & standard deviation • σ²=Σx² x P(X=x) - μ² • 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) • σ²= np(1-p) • σ=

  19. TASK / HOMEWORK • Exercise E • Mixed Questions • Test Your self

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