Intro to Probabiltiy theory notes for Elements Class.pptx

4 1 4 3 1 2 3 3 4 5 4 1 4 3 1 2 4 5 4 1 4 3 1 5 4 3 4

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4 1 4 3 1 2 3 3 4 5 4 1 4 3 1 2 4 5 4 1 4 3 1 5 4 3 4 1 4 3 Exercise Write down the first 3 terms of the Binomial Expansion of ( ⅞ + ⅛ ) 6 and evaluate their values by the use of a calculator. [ 0.4488, 0.3847, 0.1374] Probability
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58 We can now consider the Binomial Distribution Coin tossing and dice rolling have been used as a means of introducing probability. The consideration of how a coin falls can conveniently illustrate some fundamental probability theory. We must assume that we are dealing with “fair coins” i.e. not two heads or coins that are in any way biased – there is one chance in two or a Pr of 0.5 of obtaining a head and a Pr of 0.5 of obtaining a tail. The development of the Binomial Distribution cab be illustrated by the following simple example of coin tossing. Consider three coins being tossed simultaneously – There are eight possible head (H) and tail (T) combinations: Probability
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59 T T T T T H T H T T H H H T T H T H H H T H H H Now consider the probability of obtaining heads when the coins have been tossed. a) The probability of obtaining no heads i.e. Pr (no H) is one chance in eight (⅛) i.e. TTT b) The probability of obtaining 1 head i.e. Pr (1 H) is three chances in eight (⅜) i.e. TTH, THT, HTT c) The probability of obtaining 2 heads i.e. Pr (2 H) is three chances in eight (⅜) i.e. THH, HTH, HHT d) Finally the probability of obtaining 3 heads i.e. Pr (3 H) is one chance in eight (⅛) i.e. HHH i.e. We have a) Pr (no H) = ⅛ b) Pr (1 H) = ⅜ c) Pr (2 H) = ⅜ d) Pr (3 H) = ⅛ } i.e. Total Probability Pr = 1 Tossing three “fair coins” simultaneously Probability
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60 Now consider the binomial expansion of (½ + ½) Where ( ½ + ½ ) Probability of tails Probability of heads i.e. Pr of Failure i.e. Pr of Success when the coin is tossed Pr (success) + Pr (failure) = 1 3 2 2 3 3 2 1 1 2 3 1 2 3 2 1 2 1 1 2 2 3 2 1 2 1 1 3 2 1 2 1 2 1 = + + + The Pr’s agree with those found above, namely ⅛, ⅜, ⅜ and ⅛ of obtaining no heads, 1 H, 2 H and 3 H respectively when three coins are tossed simultaneously. Probability
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61 We could repeat this for the tossing of four coins. It is clear that with such probability distributions (known as binomial distributions) that the probability results can be obtained by using the binomial distribution. General Binomial Distribution In the binomial distribution of Where p = Pr of the event happening or success q = Pr of the event not happening or failure n = Number of tests Where p + q = 1 Then the terms of the binomial expansion namely: n p) (q ... , p q 2 1 1) n(n p, q 1 n , q 2 2 n 1 n n Give the respective Pr’s of obtaining 0, 1, 2, 3, etc events happening or successes.
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