Understanding the Binomial Distribution and Probability Calculations
The binomial distribution involves two possible outcomes, success or failure, in a fixed number of trials with a constant probability of success. Examples and probability-based questions illustrate how to calculate probabilities using the binomial distribution and tree diagrams.
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The Binomial Distribution Bi The Facts Bi two In the Binomial distribution there are only two possible outcomes a success or failure There is a fixed number of trials (n) Each trial has only two possible X ~ B (n,p) outcomes a success or a This means the discrete variable X failure can be represented by the Binomial The probability of a success (p) Distribution with n number of is constant from trial to trial trials, p the probability of a success. Trials are independent
Example Determine the probability of getting exactly two heads in three tosses of a biased coin for which the probability of a head = 2 3 We can solve this problem using a tree diagram easily 2 3 H 2 3 P(2 Heads) = P(HHT) + P(HTH) + P(THH) H 1 3 T 2 3 H 2 3 H P(HHT) = P(HTH) = P(THH) 2 3 3 27 1 3 2 1 =4 = T 1 3 T 2 3 P(2 Heads in 3 tosses) = 3(4 H 27) 2 3 H 1 3 1 3 = 4 T T 9 2 3 H 1 3 T 1 3 T
Probability based Questions A fair die is rolled 8 times. Find the probability of: a) No sixes b) Only 3 sixes Q Probability of not rolling a six is 5 a 6 ? 8 5 6 ? ?? ????? = b 3 were six and 5 weren t six. If S means a six was thrown and N means it wasn t throw, one possibility is 3 5 1 6 5 6 SSNNNSNN, which has probability But there s 8 3 8 3 ? 8! = 3!5!ways of arranging 3 S s and 5 N s. So: 1 6 6 3 5 5 ? 3 ????? = = 0.104
Example 2 Determine the probability of getting exactly two heads in sixty-five tosses of a biased coin for which the probability of a head = 2 3 We only use tree diagrams when n is small, so we check to see if we can use the Binomial Distribution There is a fixed number (n) of trials Each trial has two possible outcomes a success or a failure The probability of a success (p) is constant from trial to trial Trials are independent of each other Yes n = 65 Yes success heads, failure = tails Yes p = 2 3 Yes trials are independent
Example 2 Determine the probability of getting exactly two heads in sixty-five tosses of a biased coin for which the probability of a head = 2 3 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 Remember If there are exactly r success, then there are n-r failures ? ? = 65 65 2 63 = 2080 ? = ? ? Can anyone explain why?
Example 2 Determine the probability of getting exactly two heads in sixty-five tosses of a biased coin for which the probability of a head = 2 3 2. Next we calculate the probability using the following formula: ? ? ??(1 ?)? ? ? ? = ? = Where 1 p, it is the probability of failure 2 63= 8.08 x 10-28 65 2 2 3 1 3 ? ? = 2 =
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 12 ) 4 . 0 = = . 0 = . 0 . 0 4 12 ( 4 8 ( ) 4 35 65 495 35 65 P X 4 significan 3 to 235 . 0 = figures t
Quickfire Questions Show the calculation required to find the indicated probability given the distribution. ?? ? ?.?? ?.?? ? ?~? 10,0.3 ? ? = ? = ?? ? ?.?? ?.?? ? ?~? 10,0.2 ? ? = ? = ? ? ?.?? ?.?? ? ?~? 5,0.1 ? ? = ? = ? ? = ?? = ?.???? ?~? 20,0.45 ? ? ? = ? = ?.???? ?~? 20,0.45 ?
Test Your Understanding ?~? 12,1 Q1 6 What is ?(? = 2)? ? ?? ? ? ? ? ? ?? ? ? ? = ? = = ?.??? What is ?(? 1)? ? ? ? = ? ? = ? + ? ? = ? ? ? = ?.??? ?? ? ?? ? ? ? ? +?? ? = ? I have a bag of 2 red and 8 white balls. ? represents the number of red balls I chose after 5 selections (with replacement). Q2 a How is ? distributed? ? ?~? ?,?.? b Determine the probability that I chose 3 red balls. ? ? ?.???.??= ?.???? ? ? ? = ? = (If you get these quickly, go on to Exercise 1B)
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 = = + ) 1 = + = ( ) 3 ( ) 0 ( ( ) 2 P X P X P X P X 12 12 12 . 0 . 0 . 0 = . 0 + . 0 + . 0 0 12 ( ) 1 11 ( ) 2 10 ( ) 35 65 35 65 35 65 0 1 2 = 005688 . 0 + . 0 . 0 + . 0 01346 . 0 = 1 11 1 1 12 35 65 66 1225 . 0 0151
Exercise 1B The random variable ?~? 8,1 Find ? ? = 2 = ?.??? ? ? = 5 = ?.???? ? ? 1 = ?.??? A student suggests using a binomial distribution to model the following situations. Give a description of the random variable, state any assumptions that must be made and give possible values for ? and ?. 1 5 3 ? A sample of 20 bolts is checked for defects from a large batch. The production process should produce 1% of defective bolts. ?~? ??,?.?? assuming bolts being defective are independent from each other. ? a The random variable ?~? 15,2 Find ? ? = 5 = ?.????? ? ? = 10 = ?.??? ? 3 ? 4 = ?.????? 3 3 ? ? Some traffic lights have three phases: stop 48% of the time, wait or get ready 4% of the time and go 48% of the time. Assuming that you only cross a traffic light when it is in the go position, model the number of times that you have to wait or stop on a journey passing through 6 sets of traffic lights. ?~? ?,?.?? assuming lights operate independently. ? b ? A balloon manufacturer claims that 95% of his balloons will not burst when blown up. You have 20 balloons. What is the probability that none of them burst? ?.????= ?.??? ?? ??? What is the probability exactly 2 burst? = ?.??? ? 4 When Stephanie plays tennis with Tim on average one in eight of her serves is an ace . How many aces does Stephanie serve in the next 30 serves against Tim? ?~? ??,? ? assuming serves are independent and probability of an ace is constant. ? c ?
Question It is known that 80% of the seeds of particular flowers will germinate in the right conditions. If a packet of 10 seeds is purchased, find the probability that: There is a fixed number (n) of trials Each trial has two possible outcomes a success ora failure The probability of a success (p) is constant from trial to trial Trials are independent of each other a) at most two will fail to germinate. [2] b) exactly 8 will germinate. [2] Hint, you may need to think carefully about what you call a success for each question B(n,p) c) Between 3 and 6 seeds inclusive will germinate [3] Find your factor
Solution a) P(X 2) = 0.6778 Number failing to germinate ~ B(10, 0.2) B1 B1 10 8 0.880.22 = 0.3020 b) M1 A1 cao c) P(X 6) P(X < 3) = 0.1208 Between 3 and 6 inclusive don t germinate M1 M1 A1
Overview So Far These are all based on the parameters we set. ? ? ? ??? ? ? Description Name Params Outcomes Prob Func Number of trials ? 0,1, ,? 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. Binomial Distribution ? ?,? ? ? = ? ???1 ?? ? ? Probability of success in each trial ? ? ? Still to cover: The Cumulative Distribution Function ?(?) Calculating ? ? and ??? ?
