Binomial distribution KUS objectives BAT know when to

Binomial distribution • KUS objectives BAT know when to use a Binomial distribution BAT use the Binomial distribution to find probabilities and combined probabilities P(X = 0) P(X = 3) P(X = 10)

Notes 2 Conditions for a Binomial distribution to be a suitable model § A fixed number of trials § e. g. coin thrown 10 times § Two outcomes, (success or failure) § e. g. Heads or Tails § The trials are independent § You use this when you multiply the probabilities together § P(success) is a constant for each trial § P(Heads) = ½ for every throw If these conditions are satisfied then we say that the random variable X (number of successes in n trials) has a n called the ‘index’ P is the ‘parameter’

WB 11 a Probability Distributions A coin is biased so that the probability of tossing heads is 0. 3 What are the probabilities of the different combinations when tossing it 5 times? x P(x) If X is the variable the number of heads obtained, X is said to be Binomially distributed with probability p of occurring over each of n trials, written X ~ B(n, p) Note that the sum of probabilities = 1

WB 11 b Probability Distributions A coin is biased so that the probability of tossing heads is 0. 3 What are the probabilities of the different combinations when tossing it 5 times? This bar line graph shows the full probability distribution It shows a discrete distribution (not continuous) It is skewed (not symmetrical) The probabilities add up to one

WB 11 c Cumulative Probability A coin is biased so that the probability of tossing heads is 0. 3 What are the probabilities of the different combinations when tossing it 5 times? Cumulative probabilty x P(x) 0. 1681 0. 5283 1 0. 3602 2 0. 3087 3 0. 1323 4 0. 0284 5 0. 0024 0. 8370 0. 9693 0. 9977 1. 0000 If we have the full distribution we can work out the cumulative probability

Note 1 Using your Calculator fx-991 EX X ~ B(5, 0. 3) We can get this table of results for WB 11 as follows menu 4 N P 7 1: List : 5 : 0. 3 x P(x) 0 0. 1681 1 0. 3602 2 0. 3087 3 0. 1323 4 0. 0284 5 0. 0024 Input 0= 1= 2= 3= 4= 5= = Input 5= 0. 3= = Note that the results are slightly different to 4 dp – this is okay, in the old A level results cam from tables and so old exam qs will have slightly different answers

Note 2 Using your Calculator fx-991 EX X ~ B(5, 0. 3) Cumulative probabilty 0. 1681 We can get this table of results using the binomial CF function on your calcuator menu 7 N P : 5 : 0. 3 0. 8370 0. 9693 0. 9977 1: Binomial CD 1: List 0. 5283 1. 0000 Input 0= 1= 2= 3= 4= 5= = Input 5= 0. 3= = Note to look up just one result choose 2: Variable instead of 1: List

WB 12 Cumulative Probability calculations – using a calculator X ~ B(10, 0. 3) What is P(X ≤ 3)? These calculations would take a long time to do manually, especially if n were larger Use your calculator to get a table of Cumulative probabilities or just the answer you want X ~ B(10, 0. 4) Find: a) P(X≤ 3) = 0. 3822 b) P(X<8) c) P(X≥ 6) d) P(X>4) x P(X≤x) 0 0. 0060 1 0. 0464 2 0. 1673 3 0. 3822 4 0. 6331 5 0. 8337 6 0. 9452 7 0. 9877 8 0. 9983 9 0. 9999

WB 13 Cumulative Probability calculations

WB 14 Cumulative Probability calculations

WB 15 Cumulative Probability calculations – Probability Prediction A spinner is designed so that the probability it lands on red is 0. 3 Mr. G uses the spinner in a class competition. Mr G wants the probability of winning a prize to be less than 5% Each member of the class has 12 spins and the number of reds is recorded Find how many reds are needed to win a prize

Practice Ex 1 C

KUS objectives BAT know when to use a Binomial distribution BAT use the Binomial distribution to find probabilities and combined probabilities self-assess One thing learned is – One thing to improve is –

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