GCSE: Probability Dr J Frost ([email protected]) GCSE Pack References: 208-215 Last modified: 15th April 2015 GCSE Specification 208. Write probabilities using fractions, percentages or decimals 209. Compare experimental data and theoretical probabilities. Compare relative frequencies from samples of different sizes. 210. Find the probability of successive events, such as several throws of a single die. Identify different mutually exclusive outcomes and know that the sum of the probabilities of all these outcomes is 1. 211. Estimate the number of times an event will occur, given the probability and the number of trials. 212. List all outcomes for single events, and for two successive events, systematically. Use and draw sample space diagrams 213. Understand conditional probabilities. Use a tree diagram to calculate conditional probability. 214. Solve more complex problems involving combinations of outcomes. 215. Understand selection with or without replacement. Draw a probability tree diagram based on given information. RECAP: How to write probabilities

Probability of winning the UK lottery: ? 1 in 14,000,000 Odds Form ___1___ ? 14000000 Fractional Form ? 0.000000714 ? 0.0000714% Decimal Form Percentage Form Which is best in this case? RECAP: Combinatorics Combinatorics is the number of ways of arranging something. We could consider how many things could do in each slot, then multiply these numbers

together. 1 How many 5 letter English words could there theoretically be? e.g. B 26 2 I x x 26 x B 26 ? x O 26 = 26 5 How many 5 letter English words with distinct letters could there be?

S 26 3 26 L M x 25 A x 24 x U 23 ? x G 22 = 7893600

How many ways of arranging the letters in SHELF? E 5 L x 4 x F H S 3 x 2? x 1 = 5! (5 factorial) STARTER: Probability Puzzles Recall that:

In pairs/groups or otherwise, work out the probability of the following: 1 If I toss a coin twice, I see a Heads and a Tails (in either order). ? 2 If I toss a coin three times, I see a 2 Heads and 1 Tail. 5 N ? 3 In 3 throws of a coin, a Heads never follows a Tails. die in a row. NN I randomly pick a number from 1 to 4, four times,

and the values form a run of 1 to 4 in any order (e.g. 1234, 4231, ...). After shuffling a pack of cards, the cards in each suit are all together. ? NNN I have a bag of different colours of marbles and of OMG ? ? ? ? 4 Throwing three square numbers on a Seeing exactly two heads in four throws of a coin. each. Whats the probability that upon picking of them, theyre all of different colours? ?

How can we find the probability of an event? 1. We might just know! For a fair die, we know that the probability of each outcome is , by definition of it being a fair die. 2. We can do an experiment and count outcomes We could throw the dice 100 times for example, and count how many times we see each outcome. Outcome 1 2 3 4

5 6 Count 27 13 10 30 15 5 R.F. This is known as a: ? This is known as an:

Theoretical Probability Experimental Probability When we know the underlying probability of an ? event. Also known as the relative frequency , it is a probability based on observing counts. ? Check your understanding Question 1: If we flipped a (not necessarily fair) coin 10 times and saw 6 Heads, then is the true probability of getting a Head? No. It might for example be a fair coin: If we throw a fair coin 10 times we wouldnt necessarily see 5 heads. In fact we could have seen 6 heads! So the ? only provides a sensible guess for relative frequency/experimental probability the true probability of Heads, based on what weve observed. Question 2: What can we do to make the experimental

probability be as close as possible to the true (theoretical) probability of Heads? Flip the coin lots of times. I we threw a coin just twice for example and saw 0 Heads, its hard to know how unfair our coin is. But if we threw it say 1000 times and saw 200 heads, then wed have a much ? more accurate probability. The law of large events states that as the number of trials becomes large, the experimental probability becomes closer to the true probability. RECAP: Estimating counts and probabilities A spinner has the letters A, B and C on it. I spin the spinner 50 times, and see A 12 times. What is the experimental probability for P(A)? Answer: times ? Answer: ?

The probability of getting a 6 on an unfair die is 0.3. I throw the die 200 times. How many sixes might you expect to get? Test Your Understanding The table below shows the probabilities for spinning an A, B and C on a spinner. If I spin the spinner 150 times, estimate the number of Cs I will see. 0.12 0.34 C A B B B Probability A

C Outcome C A P(C) = 1 0.12 0.34 = 0.54 Estimate Cs seen?= 0.54 x 150 = 81 B I spin another spinner 120 times and see the following counts: Outcome A B C Count

30 45 45 What is the relative frequency of B? 45/120 = 0.375 ? A So far 208. Write probabilities using fractions, percentages or decimals 209. Compare experimental data and theoretical probabilities. Compare relative frequencies from samples of different sizes. 210. Find the probability of successive events, such as several throws of a single die. Identify different mutually exclusive outcomes and know that the sum of the probabilities of all these outcomes is 1.

211. Estimate the number of times an event will occur, given the probability and the number of trials. 212. List all outcomes for single events, and for two successive events, systematically. Use and draw sample space diagrams 213. Understand conditional probabilities. Use a tree diagram to calculate conditional probability. 214. Solve more complex problems involving combinations of outcomes. 215. Understand selection with or without replacement. Draw a probability tree diagram based on given information. RECAP: Events Examples of events: Throwing a 6, throwing an odd number, tossing a heads, a randomly chosen person having a height above 1.5m. The sample space is the set of all outcomes. ? An event is a description of one or more outcomes. ? It is a subset of the sample space. 1

( )= 3 We often use capital letters to represent an event, then use to mean the probability of it. The sample space 2 3 5 1 From Year 7 you should be familiar with representing sets using a Venn Diagram, although you wont need to at GCSE. Independent Events ! When a fair coin is thrown, whats the probability of:

? And when 3 fair coins are thrown: p(1st coin H and 2nd coin H and 3rd coin H) = 1 ? 8 Therefore in this particular case we found the following relationship between these probabilities: P(event1 and event2 and event3) = P(event1) x P(event2) x P(event3) ? !Mutually Exclusive Events If A and B are mutually exclusive events, they cant happen at the same time. Then: P(A or B) = P(A) + P(B) ! Independent Events If A and B are independent events, then the outcome of one doesnt affect the other. Then:

P(A and B) = P(A) P(B) But be careful 1 2 3 4 5 6 7 8 P(num divisible by 2 and by 4) = 1 ? 4

P(num divisible by 2) = 1 ? 2 P(num divisible by 4) = 1? 4 Why would it have been wrong to multiply the probabilities? Add or multiply probabilities? Getting a 6 on a die and a T on a coin. + Hitting a bullseye or a triple 20.

+ Getting a HHT or a THT after three throws of an unfair coin (presuming weve + Getting 3 on the first throw of a die and a 4 on the second. +

Barts favourite colour being red and Pablos being blue. + Shaans favourite colour being red or blue. + already worked out P(HHT) and P(THT). Independent? Event 1 Event 2 Throwing a heads

on the first flip. Throwing a heads on the second flip. No Yes It rains tomorrow. It rains the day after. No Yes That I will choose maths at A Level.

That I will choose Physics at A Level. No Yes Have a garden gnome. Being called Bart. No Yes Test Your Understanding a The probability that Kyle picks his nose today is 0.9. The probability that he independently eats cabbage in the canteen today is 0.3. Whats the probability that

Kyle picks his nose, but doesnt eat cabbage? ? b I pick two cards from the following. What is the probability the first number is a 1 and the second number a 2? 1 2 2 3 ? I throw 100 dice and 50 coins. Whats the probability I get all sixes and all heads? c ? Tree Diagrams Question: Given theres 5 red balls and 2 blue balls. Whats the probability that after two picks we have a red ball and a blue ball?

Bro Tip: Note that probabilities generally go on the lines, and events at the end. 5? 7 2 ? 7 4 ? 6 R B 2? 6 5 ? 6 1

? 6 After first pick, theres less balls to choose from, so probabilities change. R B R B Tree Diagrams Question: Give theres 5 red balls and 2 blue balls. Whats the probability that after two picks we have a red ball and a blue ball? We multiply across the matching branches, then add these values. 5 7 2 7 10 P(red and blue) = ?

21 4 6 R B 2 6 5 6 1 6 R B 5 ? 21 R

5 ? 21 B Summary ...with replacement: The item is returned before another is chosen. The probability of each event on each trial is fixed. ...without replacement: The item is not returned. Total balls decreases by 1 each time. Number of items of this type decreases by 1. Note that if the question doesnt specify which, e.g. You pick two balls from a bag, then PRESUME WITHOUT REPLACEMENT. Example (on your sheet) 3 ?8 3 8?

5 ? 8 3 ? 8 5 8? 5 5 25 ?= 8 8 64 3 5 5 3 +? 8 8 8 8 ( )( ) Question 1 1 1 1 = ?

5 5 25 ( 1 4 4 1 8 + ? = 5 5 5 5 25 1 ) ( ) 8 17 25 =

? 25 Question 2 0.9 0.9 ? 0.1 ? 0.1 ? 0.9 0.1 0.92 =0.81 ? 2 0.1 0.9=0.18 ? Question 3

5 ? 14 9 ? 14 4 13 9 13 ?5 13 8 13 two consonants? ( 9 8 36 ? = 14

13 91 5 9 9 5 45 + ? = 14 13 14 13 91 )( ) Question 4 3 ? 10 7 ?

10 2 9 7 9 ? 3 9 6 9 ( 3 7 7 3 7 + ? = 10 9 10 9 30 )( 1

) 7 23 = 30? 30 Question 5 The Birthday Paradox ? ? = . ?

? Thats surprisingly likely! Question 6 ? ? 64 110 Question 7 (Algebraic Trees) 4 4 1 ?= 4 4 16

Question 8 1 9 10 ? Question N [Maclaurin M68] I have 44 socks in my drawer, each either red or black. In the dark I randomly pick two socks, and the probability that they do not match is . How many of the 44 socks are red? Suppose there are red socks. There are therefore grey socks. ? Doing without a tree: Listing outcomes Its usually quicker to just list the outcomes rather than draw a tree. BGG:

GBG: ? Working GGB: Answer = 1904 ? 4495 Test Your Understanding Q I have a bag consisting of 6 red balls, 4 blue and 3 green. I take three balls out of the bag at random. Find the probability that the balls are the same colour. RRR: GGG: BBB: N ? Whats the probability theyre of different colours:

RGB: Each of the orderings of RGB will have the same probability. So ? Probability Past Paper Questions Provided on sheet. Remember: 1. List the possible events that match. 2. Find the probability of each (by multiplying). 3. Add them together. Past Paper Questions ? Past Paper Questions ? Past Paper Questions ?

Past Paper Questions ? 2 42 ? 16 42 Past Paper Questions 222 380 ? Past Paper Questions ? ?

64 110 Past Paper Questions ? Past Paper Questions ?