S 1 Chapter 5 Probability Dr J Frost

S 1 Chapter 5 : : Probability Dr J Frost (jfrost@tiffin. kingston. sch. uk) www. drfrostmaths. com Last modified: 17 th November 2015

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Overview In the exam you’re expected to be able to solve the following types of questions: (and I’ve checked since 2002!) #1: Enumerate matching outcomes to calculate probabilities. #2: Construct a Venn diagram (of frequencies) given information and calculate probabilities from it.

Overview In the exam you’re expected to be able to solve the following types of questions: (and I’ve checked since 2002!) #3: Construct and use Venn Diagrams involving Probabilities #4: Use laws of probability (independent, mutually exclusive and conditional events)

Overview #5: Deal with tree diagrams.

#1 : : Enumerate matching outcomes ? outcomes. A sample space is the set of all possible At GCSE you should be familiar with listing outcomes and then finding the probability of each. ? ? ?

Test Your Understanding May 2013 (R) The score S when a spinner is spun has the following probability distribution. s P(S = s) 0 0. 2 1 0. 2 2 0. 1 4 0. 3 5 0. 2 (Parts (a) to (e) we are skipping) The spinner is spun twice. The score from the first spin is S 1 and the score from the second spin is S 2. The random variables S 1 and S 2 are independent and the random variable X = S 1 × S 2. This effectively means “and” (f) Show that P({S 1 = 1} ∩ X < 5) = 0. 16. (2) (g) Find P(X < 5) (3) f ? g ?

Exercise 1 (on provided sheet) ? ?

Exercise 1 (on provided sheet) ?

Exercise 1 (on provided sheet) ? ? ?

Exercise 1 (on provided sheet) ?

#2 : : Venn Diagrams involving Frequencies ? ? Dr Frost’s cat “Pippin” ? ? ?

#2 : : Venn Diagrams involving Frequencies Conditional Probabilities Given that a randomly chosen person owns a cat, what’s the probability they own a dog? ? Dr Frost’s cat “Pippin”

Test Your Understanding Jan 2012 Q 6 The following shows the results of a survey on the types of exercise taken by a group of 100 people. 65 run 48 swim 60 cycle 40 run and swim 30 swim and cycle 35 run and cycle 25 do all three (a) Draw a Venn Diagram to represent these data. (4) Find the probability that a randomly selected person from the survey (b) takes none of these types of exercise, (2) (c) swims but does not run, (2) (d) takes at least two of these types of exercise. (2) Jason is one of the above group. Given that Jason runs, (e) find the probability that he swims but does not cycle. (3) Bro Tip: You’ll lose a mark if you don’t have a box! ? ? ? (e) Uses an alternative method which we’ll learn later when we encounter conditional probabilities.

Exercise 2 (on provided sheet) ? ?

Exercise 2 (on provided sheet) ? ? ? ?

Exercise 2 (on provided sheet) a ? b ? c ? d ? e ?

Events ? An event is a set of (one or more) outcomes. In a Venn diagram an event is represented by a circle. 3 4 1 2 6 5

Some fundamentals 3 4 1 2 6 5 What does it mean in this What is the resulting set context? of outcomes? Not A. i. e. Not rolling an even number. ? A or B. i. e. Rolling an even or prime number. ? ? A and B. i. e. Rolling a number which is even and prime. ? ? ?

Some fundamentals 3 4 1 2 6 5 What does it mean in this What is the resulting set context? of outcomes? Rolling a number which is even and not prime. ? Rolling a number which is not [even or prime]. ? Rolling a number which is not [even and prime]. ? ? ? ?

What area is indicated? A C B S ?

What area is indicated? A C B S ? or alternatively… ?

What area is indicated? A C B S ?

Probabilities in Venn Diagrams B A C ?

Test Your Understanding Construct Venn Diagrams that incorporate the following relationships between the events: Based on May 2013 Q 3 The events F, H and C are that an employee is a full-time worker, part-time worker or contractor respectively. Let B be the event that an employee uses the bus. Some full-time workers use the bus, some part-time workers use the bus and some contractors use the bus. Draw a Venn diagram to represent the events F, H, C and B. (Put a 0 in any region that won’t be used) ?

Using a Venn Diagram to Find Probabilities Click to Broculate probabilities > 0. 15 0. 6 0. 25 0 a b c ? ? ? The principle of adding/subtracting probabilities is exactly the same as when you were doing it with frequencies.

Test Your Understanding 0. 33 0. 32 ? 0. 3 0. 05 a b c d ? ? ? ?

Exercise 3 1 (on provided sheet) Given the Venn Diagram, find the total value in each region (or sketch the region). 2 16 a b c d e f g h i j 4 1 2 a 64 ? 128 32 8 ? ? ? ? ? b ? c ?

Exercise 3 3 (on provided sheet) 5 ? 6 ? ? ? 4 7 ? Jeremy and Andy are playing archery. The probability Jeremy hits the target is 0. 3. The probability neither hits the target is 0. 2. The probability both hit the target is 0. 11. Determine the probability that: (a) Andy hits the target and Jeremy doesn’t. 0. 5 (b) Exactly one of them hits the target. 0. 69 (c) Andy doesn’t hit the target. 0. 39 ? ? ?

#4 : : Laws of Probability If events A and B are mutually exclusive, then: they can’t happen at? the same time. ? ? If events A and B are independent, then: ? other. one doesn’t affect the ? But we’re interested in how we can calculate probabilities when events are not mutually exclusive, or not independent.

Addition Law Think about the areas… Mutually Exclusive ? Not Mutually Exclusive ? Known as the Addition Law.

Conditional Probability Think about how we formed a probability tree at GCSE: ? Read the ‘|’ symbol as “given that”. i. e. “B occurred given that A occurred”. Alternatively (and more commonly): ? Bro Tip: You’re dividing by the event you’re conditioning on.

Quickfire Examples ? ? ?

Full Laws of Probability ? ? In general: ? ?

Check your understanding ? ? Bro Tip: We know how do (a) and (b) from the previous exercise. Try to use Venn Diagrams in general – using probability laws when you require them. ? 0. 48 0. 11 0. 17 Click to reveal Venn Diagram 0. 24

Further Practice 1 ? ? 2 ? 3 ? ?

SUPER IMPORTANT TIPS If I were to identify one tip that will possible help you the most in probability questions, it’s this: If you see the words ‘given that’, Immediately write out the law for conditional probability. Example: “Given Bob walks to school, find the probability that he’s not late…” ? If you see the words ‘are independent’, Immediately write out the laws for independence. (Even before you’ve finished reading the question!) ? If you’re stuck on a question where you have to find a probability given others, it’s probably because you’ve failed to take into account that two events are independent or mutually exclusive, or you need to use the conditional probability or additional law.

Test Your Understanding May 2013 (R) Q 6 ? ? ?

One last very common question… ?

Exercise 4 (on provided sheet) ? ?

Exercise 4 (on provided sheet) ? ? ?

Exercise 4 (on provided sheet) ? ?

Exercise 4 (on provided sheet) ? ? ?

Exercise 4 (on provided sheet) (Parts (a) and (b) are the same as Q 3 in Exercise 3) ? ? ?

Exercise 4 (on provided sheet) a) ? ?

Exercise 4 (on provided sheet) ? ?

#5 : : Probability Trees are useful when you have later events conditioned on earlier ones, or in general when you have lots of conditional probabilities. Example: You have two bags, the first with 5 red balls and 5 blue balls, and the second with 3 red balls and 6 blue balls. You first pick a ball from the first bag, and place it in the second. You then pick a ball from the second bag. Complete the tree diagram. 1 st pick 2 nd pick ? ? ? ?

Probability Trees Key Point: When you need to find a probability using a tree, consider all possible paths in which that event is satisfied, and add the probabilities together. ? ? ?

Check your understanding Of 120 competitors in a golf tournament, 68 reached the green with their tee shot on the first hole. Of these, 46 completed the hole in 3 shots or less. In total, 49 players took more than 3 shots on the first hole. Click to reveal Tree Diagram ? ? ?

Exercise 5 (on provided sheet) ?

Exercises (on worksheet) 2 ? 23 ? 5 12 ? 7 12 ? 13 ? 12 b P(H) = (5/12 x 2/3) + ? (7/12 x ½) = 41/72 c P(R|H) = P(R n H) / P(H) = (5/18) / (41/72) ? = 20/41 d P(RR or BB) = (5/12)2 + ? (7/12)2 = 37/72

Exercise 5 (on provided sheet) ? ? ? ?

Exercise 5 (on provided sheet) ? ? ?

To Finish Off : : A Classic Conundrum I have two children. One of them is a boy. What is the probability the other is a boy? ? METHOD 1 The ‘restricted sample space’ method BB BG GB GG There’s four possibilities for the sex of the two ? children, but only 3 match the description. In 1 out of the 3 possibilities METHOD 2 Using conditional probability ?