Probability

Probability This Chapter is on Probability We will review methods used on GCSE Maths There will be new ideas introduced many of which will require diagrams to help solve Diagrams will include Probability Trees, Sample Spaces and Venn Diagrams

Teachings for Exercise 5A

Probability Probability Reminder An experiment is a repeatable process that gives outcomes An event is a collection of one (or more) outcomes A sample space is the set of all possible outcomes in an experiment P(event occurring) = Impossible event  P = 0 Certain event  P = 1 Probabilities are only written as Fractions, Decimals and Percentages. No. outcomes in that event Total possible outcomes 5A

Probability Probability Reminder Find the probability of a fair dice landing on a 5. P(event occurring) = P(5) = No. outcomes in that event Total possible outcomes 1 6 5A

Probability Probability Reminder Two spinners are numbered 1-4. Both are spun and the sum of the numbers calculated. Find P(5) and P(x > 5) 5 6 7 8 4 Draw a sample space to show the outcomes. P(5) = P(x > 5) = 4 5 6 7 3 Spinner 2 4 1 = 3 4 5 6 2 16 4 2 3 4 5 1 6 3 1 2 3 4 = 16 8 Spinner 1 5A

Teachings for Exercise 5B

Probability Using Venn Diagrams Venn diagrams are a very useful way of representing Probabilities. They can also help you answer multi-part questions. S A B A rectangle labelled S represents the Sample Space Circle A represents the Probability of event A Circle B represents the Probability of event B 5B

The area outside of A represents the Probability of A not happening. Using Venn Diagrams A B The Area in the middle represents the Probability of A and B happening together. ‘n’  ‘and’ S The whole area represents the Probability of A or B happening (or them together). A B S The area outside of A represents the Probability of A not happening. A B 5B

Probability Using Venn Diagrams A card is selected at random from a pack of 52 playing cards. Let A be the event that the card is an ace, and D be the event that the card is a diamond. Draw a Venn diagram to show this information. Always fill in the middle first. The middle represents an ace and a diamond.  1 card There are 4 aces in total, one of which has already been filled in  3 cards extra in ‘A’ There are 13 diamonds, one of which has been filled in  12 extra cards in ‘D’ 4) 52 cards in total, subtract the 16 that have been used  36 cards left outside the circles S A D 3 12 1 36 5B

Probability Using Venn Diagrams A card is selected at random from a pack of 52 playing cards. Let A be the event that the card is an ace, and D be the event that the card is a diamond. Draw a Venn diagram to show this information. 1 52 S ‘Probability of an Ace and a Diamond’ 16 52 4 13 A D ‘Probability of an Ace or a Diamond’ 3 12 1 48 52 12 13 ‘Probability of it not being an Ace’ 36 12 52 3 13 ‘Probability of it not being an Ace, and being a Diamond’ 5B

Probability Using Venn Diagrams In a class of 30 students, 7 are in the choir, 5 are in the school band and 2 are in both the choir and the band. Draw a Venn diagram to show this information. Always fill in the middle first. The middle represents choir and band.  2 students There are 5 students in the band, in total. 2 are already on the diagram.  3 students extra in ‘B’ There are 7 students in the choir, 2 of which are already on the diagram.  5 more students in ‘C’ 4) 30 students in total, 10 already filled in.  20 students outside the circles S B C 3 5 2 20 5B

Probability Using Venn Diagrams In a class of 30 students, 7 are in the choir, 5 are in the school band and 2 are in both the choir and the band. Draw a Venn diagram to show this information. ‘Probability of not being in the band’ S B C 3 5 2 20 You could also have got 25/30 by counting the parts not in the ‘B’ circle. 5B

Probability Using Venn Diagrams A vet surveys 100 clients. She finds out the following: 25 have dogs 53 have cats 40 have fish 15 have dogs and cats 10 have cats and fish 11 have dogs and fish 7 have dogs, cats and fish S Always fill in the middle first. The middle represents all 3 pets D C 6 8 35 2) Then fill in the parts where 2 circles overlap. Remember to take away the middle from each. 7 4 3 3) After this you can fill in the rest, based on what you have already completed 26 4) Remember to work out how many people have no pets (add up the numbers in the circle, and subtract from 100) F 11 5B

Probability Using Venn Diagrams A vet surveys 100 clients. She finds out the following: 25 have dogs 53 have cats 40 have fish 15 have dogs and cats 10 have cats and fish 11 have dogs and fish 7 have dogs, cats and fish S 6 100 3 50 D C 6 8 35 60 100 3 5 7 4 3 11 100 26 F 11 5B

Teachings for Exercise 5C

Probability Formulae from the Venn Diagram – The Addition Rule S A B If P(A) = a and P(B) = b S And we let the intersection = i Then we can label a Venn diagram as follows: A B The intersection  i The rest of A  a – i The rest of B  b – i The Area in the circles  (a – i) + i + (b – i)  a + b – i So the remainder will be 1 – (a + b – i) a - i i b - i 1 – (a + b – i) 5C

Rearranged you can also get this formula Probability Formulae from the Venn Diagram – The Addition Rule If P(A) = a and P(B) = b S And we let the intersection = i A B The Probability of A or B is the whole of the area inside the circles P(A or B) = (a – i) + (b – i) + i P(A or B) = a – i + b – i + i P(A or B) = a + b - i a - i i b - i 1 – (a + b – i) P(A or B) = a + b - i Rearranged you can also get this formula 5C

Probability Formulae from the Venn Diagram – The Addition Rule A and B are two events such that P(A) = 0.6, P(B) = 0.7 and P(A or B) = 0.9. Calculate: a) b) c) d) a) S A B 0.2 0.4 0.3 0.1 Now you know the intersection, you can draw a Venn diagram! 5C

Probability Formulae from the Venn Diagram – The Addition Rule A and B are two events such that P(A) = 0.6, P(B) = 0.7 and P(A or B) = 0.9. Calculate: b) c) d) b) S A B 0.2 0.4 0.3 ‘Probability of not A’ 0.1 ‘Probability of not A, or B’ ‘Probability of not A, and B’ 5C

Teachings for Exercise 5D

Probability Formulae from the Venn Diagram – The Multiplication Rule Conditional Probability is where the probability of an event is affected by whether another event has already occurred or not.  For example, the probability of choosing an ace from a pack of cards will be affected if a random card has been removed We have learnt the ‘addition’ rule, and now we will learn the ‘multiplication rule’  The multiplication rule will allow us to solve problems involving conditional probability 5D

Probability Formulae from the Venn Diagram – The Multiplication Rule 4 Two fair spinners are both numbered 1-4. They are thrown together and the sum recorded. Given that at least one spinner lands on a 3, find the probability of the spinners indicating a sum of 5. Using the sample space: We only consider the results that have a 3 as one of the numbers (there are 7 possibilities here) Out of these, 2 have a sum of 5 So P(sum of 5 | at least one 3) 2/7 5 6 7 8 4 4 5 6 7 3 Spinner 2 3 4 5 6 2 2 3 4 5 1 1 2 3 4 Spinner 1 5D

Probability Formulae from the Venn Diagram – The Multiplication Rule A We are going to work out the formula for the probability of event B, given that event A has happened. We are saying that A has happened, and so are only including the ‘A’ area A B a - i i b - i As A has happened, the ‘A’ area represents all the possibilities ‘i’ is the part of the circle where B has happened So the probability B happens, given A has: The vertical line means ‘given’ 5D

Probability Formulae from the Venn Diagram – The Multiplication Rule A We are going to work out the formula for the probability of event B, given that event A has happened. A B a - i i b - i Re-arranging, we get… OR 5D

The probabilities must add up to 1 Probability Formulae from the Venn Diagram – The Multiplication Rule C and D are two events such that P(C) = 0.2, P(D) = 0.6 and P(C|D) = 0.3. Find: a) b) c) P(D) = 0.6, so subtract 0.18 We will construct a Venn Diagram to help… We need the probability of the intersection: C D 0.02 0.18 0.42 0.38 P(C) = 0.2, so subtract 0.18 The probabilities must add up to 1 5D

Probability Formulae from the Venn Diagram – The Multiplication Rule C and D are two events such that P(C) = 0.2, P(D) = 0.6 and P(C|D) = 0.3. Find: a) b) c) a) C D 0.02 0.18 0.42 0.38 5D

Probability Formulae from the Venn Diagram – The Multiplication Rule C and D are two events such that P(C) = 0.2, P(D) = 0.6 and P(C|D) = 0.3. Find: a) b) c) 0.9 0.38 0.42 b) C D ‘Probability of not C and not D’ 0.02 0.18 0.42 c) ‘Probability of not C and D’ 0.38 5D

Denominators must be the same Probability Formulae from the Venn Diagram – The Multiplication Rule Let A and B be events such that P(A) = 3/10, P(B) = 2/5 and = 1/2. Find: a) b) P(B) = 2/5 in total We will construct a Venn Diagram to help… We need the probability of the intersection: 2/5 – 1/5 A B 1 10 1 5 1 5 Denominators must be the same 1 2 P(A) = 3/10 in total 1 - (1/10 + 1/5 + 1/5) 1 - (1/10 + 2/10 + 2/10) 3/10 – 1/5 1 - (5/10) 3/10 – 2/10 5D

Probability Formulae from the Venn Diagram – The Multiplication Rule Let A and B be events such that P(A) = 3/10, P(B) = 2/5 and = 1/2. Find: a) b) 2 3 a) A B 1 10 1 5 1 5 1 2 5D

Probability Formulae from the Venn Diagram – The Multiplication Rule Let A and B be events such that P(A) = 3/10, P(B) = 2/5 and = 1/2. Find: a) b) 2 3 b) A B 1 10 1 5 1 5 1 2 P(A’) = 1 – P(A) = 1 – 3/10 5D

Teachings for Exercise 5E

Probability Tree Diagrams You will have seen Tree Diagrams at GCSE level, and they can also be used to represent conditional probabilities. The number of spectators at an event is dependent on the weather. On a rainy day, the probability of a big turnout is 0.4. However, if it does not rain, there is a probability of 0.9 that there will be a big turnout. The weather forecast gives a 0.75 probability of rain. Show this on a tree diagram. The rain is the independent event (ie – it affects the probability of the other), so it comes first. P(R) = 0.75 P(R’) = 0.25 B 0.4 R 0.75 B’ 0.6 B 0.9 The second set of possibilities are a high turnout (B), or not (B’)  The probabilities are different depending on whether it rained or not… 0.25 R’ 0.1 B’ 5E

Probability B R B’ B R’ B’ 5E Tree Diagrams 0.4 0.75 0.6 0.9 0.25 0.1 You can use the Multiplication rule to work out probabilities. ‘Probability of a big turnout and rain’ 5E

Probability B R B’ B R’ B’ 5E Tree Diagrams 0.4 0.75 0.6 0.9 0.25 0.1 You can use the Multiplication rule to work out probabilities. ‘Probability of not a big turnout and rain’ 5E

Probability B R B’ B R’ B’ 5E Tree Diagrams 0.4 0.75 0.6 0.9 0.25 0.1 You can use the Multiplication rule to work out probabilities. ‘Probability of a big turnout and not rain’ 5E

Basically, remember to multiply along each ‘path’ Probability Tree Diagrams B 0.4 Checking… 0.3 + 0.45 + 0.225 + 0.025 1 R 0.75 B’ 0.6 B 0.9 0.25 R’ 0.1 B’ You can use the Multiplication rule to work out probabilities. ‘Probability of not a big turnout and not rain’ Basically, remember to multiply along each ‘path’ 5E

There are 2 ways of having a big turnout Probability Tree Diagrams B 0.4 R 0.75 B’ 0.6 B 0.9 0.25 R’ 0.1 B’ Calculate the probability of a big turnout. There are 2 ways of having a big turnout 5E

Probability Tree Diagrams A bag contains 7 green beads and 5 blue beads. A bead is taken at random, the colour recorded and the bead is not replaced. A second is then taken and the colour recorded. Find P(1 Green and 1 Blue). One less Green 6 11 There is the possibility of Green or Blue both times. P(G1) = 7/12 P(B1) = 5/12 Blue the same as to begin with G2 G1 7 12 5 11 B2 7 11 G2 5 12 The second set of possibilities depend on what colour was taken the first time. There will be 11 left, and one less of either Green or Blue. B1 4 11 B2 Green the same as to begin with One less Blue 5E

Probability Tree Diagrams A bag contains 7 green beads and 5 blue beads. A bead is taken at random, the colour recorded and the bead is not replaced. A second is then taken and the colour recorded. Find P(1 Green and 1 Blue). 6 11 As we want one of each, there are 2 possible routes: G2 G1 7 12     7 12 5 11 35 132 5 11 B2 x = 7 11 5 12 7 11 35 132 G2 x = 5 12 B1 4 11 B2 5E

Teachings for Exercise 5F

The Addition Rule for Mutually exclusive Events Probability S Mutually Exclusive and Independent Events When 2 events cannot happen at the same time, they are Mutually Exclusive.  If we apply this to the Addition Rule: You can also work backwards. If the above is true then the events are Mutually Exclusive. A B The Addition Rule for Mutually exclusive Events 5F

The Multiplication Rule for Independent Events Probability Mutually Exclusive and Independent Events When one event has no effect on another, they are said to be independent. So the Probability of A, given B, is just the same as the probability of A on its own. Applying this to the Multiplication Rule. Again, you can work backwards. If you put the numbers you are given into the above formula, and it works, then the events are independent. The Multiplication Rule for Independent Events 5F

Probability Mutually Exclusive and Independent Events Events A and B are Mutually Exclusive and P(A) = 0.2 and P(B) = 0.4 Calculate: a) b) c) Construct a Venn Diagram S A B 0.2 0.4 0.4 Mutually Exclusive, so the circles are separate 5F

Construct a Venn Diagram Probability Mutually Exclusive and Independent Events Events C and D are Independent and P(C) = 1/3 and P(D) = 1/5 Calculate: a) b) c) Construct a Venn Diagram S C D 4/15 1/15 2/15 8/15 P(C) = 1/3 in total so: P(D) = 1/3 in total so: 5F

Summary We have now finished all the topics for Probability You must remember both the Addition Rule and the Multiplication Rule. If you aren’t sure, think ‘what am I trying to find out?’ The best way to solve a problem is to draw a diagram to help, when you have enough information to do so You also need to remember how the rules vary for Mutually Exclusive and Independent Events