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This text discusses probability calculations in a math book and the probability of drawing counters from a bag. It also explains the expected number of attempts to draw green counters and the mean and variance of surviving gum trees.

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  1. STARTER The probability that a particular page in a maths book has a misprint is 0.2. Find the probability that of 12 pages in the book: • 4 of them have a misprint n = 12 and p = 0.2 P(X = 4) = 0.133 • Fewer than 2 of them contain a misprint. P(X < 2) = 0.0687 + 0.2062 = 0.275

  2. STARTER A bag contains 2 red counters and 3 green counters. Two counters are drawn at random from the bag without replacement. a) Find the probability that exactly 2 green are drawn • Find the probability that out of 15 attempts of drawing two counters (replaced after each 2 are drawn) more than 12 of them will result in exactly 2 green counters been drawn • What is the expected number of attempts in which exactly 2 green counters will be drawn 3/5 x 2/4 = 3/10 P = success of 2 green drawn = 3/10 Then X B(15, 3/10) P(X>12) = 8.72x 10-6 15 x 3/10 = 4.5

  3. Note 6: Variance of a Binomial Distribution For the binomial distribution where the Variance (a measure of dispersion) of X is Var(X) = npq where q = 1 – p Standard Deviation = √Var(X)

  4. Example: A farmer plants 150 gum trees, each of which have an individual probability of surviving the winter conditions of 0.8. Find the mean and variance of the number that survive the winter. Mean μ = np Variance Var(X) = npq = 150 × 0.8 = 120 = 150 × 0.8 × 0.2 = 24

  5. New Book Page 535 Exercise 15G

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