The Binomial Distribution Bi The Facts Bi two

The Binomial Distribution Bi The Facts Bi – two In the Binomial distribution there are only two possible outcomes – a “success” or “failure” v There is a fixed number of trials X ~ B (n, p) v Each trial has only two possible This means the discrete variable X outcomes – a “success” or a can be represented by the Binomial “failure” Distribution with n number of trials, p the probability of a success. (n ) v The probability of a success (p) is constant from trial to trial v Trials are independent

Example We can solve this problem using a tree diagram easily … H H H H T T T H H T T T H T

Probability based Questions Q A fair die is rolled 8 times. Find the probability of: a) No sixes b) Only 3 sixes a ? b ?

Example 2 We only use tree diagrams when n is small, so we check to see if we can use the Binomial Distribution v There is a fixed number (n) Yes – n = 65 of trials v Each trial has two possible outcomes – a “success” or a “failure” Yes – success – heads, failure = tails v The probability of a success (p) is constant from trial to trial v Trials are independent of each other Yes – trials are independent

Example 2 1. First we need to figure out how many combinations there are: This represents the number of ways of choosing r items out of n items Can anyone explain why?

Example 2 2. Next we calculate the probability using the following formula: Where 1 – p, it is the probability of failure

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

Quickfire Questions Show the calculation required to find the indicated probability given the distribution. ? ? ?

Test Your Understanding Q 1 ? ? Q 2 a b ? ? (If you get these quickly, go on to Exercise 1 B)

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

Exercise 1 B 1 5 ? a 3 ? ? ? b ? 4 ? ? ? c ?

Question It is known that 80% of the seeds of particular flowers will germinate in the right conditions. v There is a fixed number If a packet of 10 seeds is purchased, find the probability that: (n) of trials v Each trial has two a) at most two will fail to germinate. [2] possible outcomes – a “success” or a “failure” b) exactly 8 will germinate. [2] v The probability of a Hint, you may need to c) Between 3 and 6 seeds inclusive will germinate [3] success (p) is constant think carefully about what from trial to trial you call a success for each v question B(n, p) Trials are independent of each other Find your factor

Solution

Overview So Far These are all based on the parameters we set. Description Name We count the number of ‘successes’ after a number of trials, each with two outcomes (‘success’ and ‘failure’). e. g. Number of heads after 10 throws of an unfair coin. Params ? Outcomes ? Prob Func ? ? ?