S 2 Chapter 1 Binomial Distribution Dr J

S 2 Chapter 1: Binomial Distribution Dr J Frost (jfrost@tiffin. kingston. sch. uk) www. drfrostmaths. com Last modified: 30 th August 2015

Probability Distributions In S 1 we saw that a random variable has an associated probability distribution. This had two components: 1 Outcomes expressed as a set 2 A probability distribution which maps outcomes to probabilities Description Throwing a fair die. Number of heads seen after throwing a coin twice. Outcomes Probability Function ? ?

Off-The-Shelf Probability Distributions These are all based on the parameters we set. Description A Bell-shaped distribution around some known mean with a known variance. Each discrete outcome is equally likely to happen. Name ? Discrete Uniform Distribution ? Params Outcomes Prob Func ? ? ? (No need to copy this – We will keep expanding this table and coming back to it)

Off-The-Shelf Probability Distributions In Chapters 1, 2, and 4, we will gradually explore three new ‘off-the-shelf’ distributions. Don’t worry yet! These are all based on the parameters we set. Description Name A Bell-shaped distribution around some known mean with a known variance. Each discrete outcome is equally likely to happen. 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. Counting the number of events which occur within a fixed time, given some known rate. Discrete Uniform Distribution Params Outcomes Prob Func

Frost True Stories Back in 2010 I was on holiday in Hawaii and visited the family of a friend. We noticed that at the dinner table that out of the 8 of us, 6 of us were left-handed (including myself). One of them asked, “The chances of that must be very low”. I saw that as a challenge. We’ll solve this problem in a sec…

S 2 – Chapter 2 – Binomial Distribution Lesson 1: Introduction to Combinatorics

Factorial and Choose Function Q 1 How many ways are there of arranging 5 different coloured beads in a line? ?

Factorial and Choose Function Q 2 How many ways are there of arranging 2 red beads and 3 blue beads in a line? We can’t distinguish each of the red beads, nor the blue beads. These 2 arrangements are actually the same! Click for Bromanimation Answer

Test Your Understanding T 1 T 4 I throw a coin 8 times. How many possibilities are there in which I threw 4 Heads? ? T 2 ? T 5 ? T 3 ? If you have 8 people at a dinner table, how many ways are there for 6 of them to be left-handed? ?

Exercise 1 X (Not in textbook) 5 1 ? ? 2 ? 3 6 ? 7 ? 4 ? 8 ? ?

Probability based Questions Q A fair die is rolled 8 times. Find the probability of: a) No sixes b) Only 3 sixes c) 4 twos and 4 sixes a ? b ? c ?

S 2 – Chapter 2 – Binomial Distribution Lesson 2: Binomial Distribution

Frost True Stories Back in 2010 I was on holiday in Hawaii and visited the family of a friend. We noticed that at the dinner table that out of the 8 of us, 6 of us were left-handed (including myself). One of them asked, “The chances of that must be very low”. I saw that as a challenge. Consider the 8 people in a line. Suppose 10% of the population is left-handed. Can you now work out the probability of 6 being left-handed now? ?

Test Your Understanding Q ? ? ?

Binomial Distribution ! So that we’re allowed to multiply the probabilities together from each trial We’re counting heads, so throwing a heads is the “success”. FICT

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

Is it Binomially Distributed? Is a Binomial Distribution appropriate as a model? Some number out of 8 people being left-handed The number of red balls selected when 3 balls are drawn from bag of 15 white and 5 red balls. Number of throws on die until 6 obtained Number of girls in family of 4 children No, not fixed. This is known as a ‘Geometric Distribution’ (which we won’t cover) ? ? ? Usually. But in my story, genetics has an influence on handedness. Technically the probability of having a girl increases if you previously had a girl, and vice versa. But the probability is still close to 0. 5, so Binomial Distribution is appropriate. ? Only if balls drawn with replacement.

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

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

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 ? ? ?

S 2 – Chapter 2 – Binomial Distribution Lesson 3: Cumulative Distribution Function

Cumulative Distribution Function ? ? ?

Cumulative Distribution Function ? ? ? Quickfire Questions ? ? ? ? ?

Test Your Understanding Q a b ? ? c ?

Exercise 1 C 1 8 ? ? a ? b ? 3 ? Further Question (not in book) ? a ? b ? You can start on your homework questions if you’ve done all of these (ask for sheet).

S 2 – Chapter 2 – Binomial Distribution Lesson 4: Probability Ranges

Dealing with Probability Ranges Q STEP 1: Represent the sentence using probability. ? STEP 3: Rearrange. STEP 4: Use table backwards to find value corresponding to closest probability.

Test Your Understanding Q At Camford University, students have 20 exams at the end of the year. All students pass each individual exam with probability 0. 45. Students are only allowed to continue into the next year if they pass some minimum of exams out of the 20. What do the university administrators set this minimum number such that the probability of continuing to next year is at least 90%? ? This is exactly what you should write.

Exercise 1 C (again) B 7 ? ? 8 ? ? ? Further Questions (not in book) A ? ?

S 2 – Chapter 2 – Binomial Distribution

Examples Q ? ? Q ?

Exercise 1 D

? Hint: Perhaps rethink the problem in terms of who doesn’t buy tea.

Quickfire Questions ? ? ?

Test Your Understanding ? S 2 May 2012 Q 8 (Part (b) was very challenging according to the Examiner’s Report!) ?

And more general probability questions… ? ?

Exercises 1 4 a ? 2 ? ? b ? 5 3 ? ?

S 2 May 2013 Q 7 ? ? ? You won’t be able to do (d) until Chapter 2.

S 2 Jan 2013 Q 3 ?

Exercise 1 C Further Exercises (not in textbook) 1 ? 2 ? 3 ?

Summary These are all based on the parameters we set. Description 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. Name Params ? Outcomes ? Prob Func ? ? ?