Probability

Understand the following

Consider each of the following statements Declare how you would best define the probability that each will happen using one of the following words Unlikely Impossible Likely Certain Even Chance

What are the chances you will watch TV when you go home tonight?

it will get dark tonight? What are the chances it will get dark tonight?

What are the chances you will meet a dinosaur on the way home today?

What are the chances you will win the next time you buy a scratchcard?

 

 

The probability scale Probability is a numerical measure of how likely or unlikely an event is to occur. Probabilities are usually written as fractions, but can be written in any form equivalent to that fraction. Eg ¾ = 0.75 = 75% Probabilities can be anywhere between 0 (impossible) and 1 (certain): Impossible Unlikely Even chance Likely Certain c b d a ½ 1 a) an event with a probability of 0.8 would be described as very likely b) an event with a probability of 0.4 would be described as unlikely c) an event with a probability of 1/20 would be described as very unlikely d) an event with a probability of 6/12 would be described as even chance

The probability scale Key words: Even chance Likely Impossible Certain Unlikely 1. Complete this probability scale using the key words given Even chance Impossible Unlikely Likely Certain ½ 1 2. Label the events described below on the probability scale: The chance of getting an even number when rolling a dice The chance of winning the National Lottery The chance of rain in March b a c ½ 1 3.Describe an event that you think has a probability of: a) 0.3 _________________________________________________ b) 1 _________________________________________________ c) 0.8 _________________________________________________

Probability of an event 1. Bob is picking randomly from a bag containing tiles numbered 1 to 10. Write down the probability that the number he picks is: a) 7 b) 4 or less c) Odd d) A multiple of 3 2. A survey is conducted of pupils’ favourite team: Team Spurs Man Utd Liverpool Arsenal Pupils 12 8 4 6 John picks a pupil at random to ask more questions. Write down the probability that the pupil he picks supports: a) Liverpool b) A London team c) Not Liverpool 3. A bag contains 20 coloured balls, some red and some blue. Keith knows that the probability of picking a red ball is 2/5. How many red balls are there? 8 red balls

3. The table shows information about the number of goals scored by Aston Villa in each game of the season so far: Mr Walker has all the games on DVD and decides to watch one. He picks a game randomly. What is the probability he picks a game with: a) exactly 1 goal by Aston Villa, Number of goals Number of games 3 1 6 2 5 4 More than 3 b) 2 or more goals by Aston Villa Total = 20 games 4. The cumulative frequency curve below shows the distribution of the height of 50 students. Estimate the probability that a student picked at random will be more than 164cm tall

Total probability of events If one of a set of possible outcomes xi must happen, then ΣP(xi) = 1 The sum of their probabilities is 1 Eg the probability of rain today is 0.7 so the probability of no rain is 0.3 Eg a coin is biased so that the probability of heads is 3/5 so the probability of tails is 2/5 Eg a 4-sided spinner has the following probabilities of getting each number. The probability that the spinner will land on 2 is equal to the probability it will land on 4. Complete the table. Number 1 2 3 4 Probability 0.2 0.3 x x

Total probability of events 1. A die is biased so that the probability of rolling a six is . What is the probability of not rolling a six? 2. A die is biased so that the probability of each number is: Find the value of x Number 1 2 3 4 5 6 Probability 0.1 0.15 x 0.2 0.05 ΣP(xi) = 1 3. The weatherman claims that it is twice as likely to snow as not. Complete the table: Snow No snow Probability

At the races Each horse moves 1 square if you get its total when you roll both dice. Is it a fair race? 2 3 4 5 6 7 8 9 10 11 12

Why isn’t the race fair? Consider the possible outcomes from finding the total of 2 dice: Dice A Total Probability 2 1 2 3 4 5 6 3 2 3 4 5 6 7 4 3 4 5 6 7 8 5 4 5 6 7 8 9 Dice B 6 5 6 7 8 9 10 7 6 7 8 9 10 11 8 7 8 9 10 11 12 9 10 11 12 7 is the most likely total, so horse 7 is most likely to win 2 & 12 are the least likely totals, so horses 2 and 12 are least likely to win

Probability using tables Eg in a game, two fair dice are rolled and a score is found by multiplying the numbers obtained together. Show the possible outcomes in the table below Use your completed table to find the probability of getting a score of 12 Use the table to find the probability of getting a score of 23 or more Dice A a) 1 2 3 4 5 6 b) 4 outcomes out of 36 give a score of 12 1 2 3 4 5 6 2 4 6 8 10 12 3 6 9 12 15 18 Dice B 4 8 12 16 20 24 c) 10 15 6 outcomes out of 36 give a score of 23+ 5 20 25 30 6 12 18 24 30 36

Probability using tables 1. In a game, two fair spinners are spun and a score is found by adding the numbers obtained together. Show the possible outcomes in the table below Use your table to find the probability of getting a score of 7 Find the probability of getting a score of 4 or less Spinner A b) 2 outcomes out of 16 give a score of 7 a) 1 2 3 4 2 3 4 5 3 4 5 6 Spinner B 4 5 6 7 c) 5 6 7 8 6 outcomes out of 16 give a score of 4 or less

2. Keith has 3 coloured balls in a bag- red, blue and yellow. He picks one, records its colour, puts it back and picks another. Complete the table to show the possible outcomes Write down the probability Keith picks: two balls of the same colour two balls of different colour at least one yellow ball bi) 3 outcomes out of 9 give the same colour 1st pick a) R B Y ii) Either they are the same colour or not RR BR YR 2nd pick RB BB YB RY BY YY iii) 5 outcomes out of 9 have a yellow ball

Tree diagrams Sometimes, a tree diagram can help you understand probabilities Eg a coin is biased so that the probability of throwing heads each time is 2/3 The branches show the possible outcomes and their probabilities Any chain of branches from the beginning to the end represents a combination of outcomes 1st throw 2nd throw H Two heads in a row H T Heads followed by tails H Tails followed by heads T T Two tails in a row

In Probability And means X Or means +

Tree diagrams Eg Johnny has a 0.4 chance of scoring from a free-kick and a 0.7 chance of scoring from a penalty The branches show the possible outcomes and their probabilities Any chain of branches from the beginning to the end represents a combination of outcomes Free-kick Penalty Score Scores both Score Miss Scores free-kick but misses penalty Score Misses free-kick but scores penalty Miss Miss Misses both

Finding probabilities with tree diagrams Eg a coin is biased so that the probability of throwing heads each time is 2/3 To find the probability of a combination of outcomes, multiply the probabilities along the relevant branches 1st throw 2nd throw H P(Two heads in a row) H T P(Heads followed by tails) H P(Tails followed by heads) T T P(Two tails in a row) If more than one combination gives the desired outcome, add their probabilities P(one head, one tail)

Finding probabilities with tree diagrams Eg Johnny has a 0.4 chance of scoring from a free-kick and a 0.7 chance of scoring from a penalty To find the probability of a combination of outcomes, multiply the probabilities along the relevant branches Freekick Penalty Score P(scores both) Score Miss P(scores one) Score Miss Miss P(scores neither) If more than one combination gives the desired outcome, add their probabilities

Tree diagrams 1. Simon plays one game of tennis and one game of snooker. The probability that Simon will win at snooker is The probability that Simon will win at tennis is a) Complete the tree diagram b) Work out the probability that Simon wins both games. c) Work out the probability that Simon will win only one game.

2. Julie and Pat are going to the cinema. The probability that Julie will arrive late is 0.2 The probability that Pat will arrive late is 0.6 The two events are independent. a) Complete the diagram. b) Work out the probability that Julie and Pat will both arrive late. c) Work out the probability that neither of them arrive late.

3. Julie throws a fair red dice once and a fair blue dice once. Complete the probability tree diagram to show the outcomes. Label clearly the branches of the probability tree diagram. The probability tree diagram has been started in the space below. b) Calculate the probability that Julie gets at least one six. = 1 – P(no sixes)

4. Loren has two bags. The first bag contains 3 red counters and 2 blue counters. The second bag contains 2 red counters and 5 blue counters. Loren takes one counter at random from each bag. a) Complete the probability tree diagram. b) Work out the probability that Loren takes one counter of each colour.

1 2 3 4 5 Play your cards right 6 7 8 9 10

Play your cards right Higher or lower??? 6 2 4 1 7 8 9 1 2 3 4 5 6 7 8 10 6 2 4 1 7 8 9 Higher or lower??? The probability of higher or lower is conditional on the cards that have already appeared

Non-replacement Eg a bag contains 3 red and 7 blue balls. A ball is picked, not replaced, and another picked. Complete the tree diagram There are 10 balls to choose from when picking the 1st ball If the object is not replaced, this affects the probabilities on the 2nd pick 1st pick 2nd pick If a red ball was picked first, there are only 2 red balls left Red Red If a red ball was picked first, there are still 7 blue balls left Blue If a blue ball was picked first, there are still 3 red balls left Red Blue If a blue ball was picked first, there are only 6 blue balls left Blue

Non-replacement Eg a bag contains 3 red and 7 blue balls. A ball is picked, not replaced, and another picked. Find the probability that: 2 red balls are picked 1 of each colour is picked To find the probability of a combination of outcomes, multiply the probabilities along the relevant branches 1st pick 2nd pick P(both red) Red Red Blue P(one of each) Red Blue If more than one combination gives the desired outcome, add their probabilities Blue

Non-replacement 1. A bag of sweets contains 2 toffees and 5 chocolates. A sweet is picked, eaten, and another picked. Complete the tree diagram Find the probability that: i) Both sweets are toffees ii) 1 of each sweet is picked 2nd pick bi) P(both toffee) 1st pick Toffee Toffee Chocolate bii) P(one of each) Toffee Chocolate Chocolate

2. 5 white socks and 3 black socks are in a drawer. Stefan takes out two socks at random. Work out the probability that Stefan takes out two socks of the same colour. 1st pick 2nd pick White P(both white) White P(both black) Black P(same colour) White Black Black

Listing outcomes systematically Eg A bag contains 3 blue beads, 5 yellow beads and 2 green beads. Sid takes a bead at random from the bag, records its colour and replaces it. He does this two more times. Work out the probability that, of the three beads Sid takes, exactly two are blue. A tree diagram would take too long here Combinations with exactly 2 blue: P(BBY) BBY and P(BBY) = P(BYB) = P(YBB) BYB YBB P(BBG) BBG and P(BBG) = P(BGB) = P(GBB) BGB GBB So P(exactly 2 blue)

Listing outcomes systematically 1. For any match, the probabilities of each result for Aston Villa are as follows: P(win) = P(draw) = P(lose) = Find the probability that, in 3 matches, Aston Villa win exactly 2 matches Combinations with exactly 2 wins: P(WWD) WWD and P(WWD) = P(WDW) = P(DWW) WDW DWW P(WWL) WWL and P(WWL) = P(WLW) = P(LWW) WLW LWW So P(exactly 2 wins)

2. A bag contains 2 blue balls and 3 green balls 2. A bag contains 2 blue balls and 3 green balls. Pete takes a ball at random from the bag, records its colour and replaces it. He does this two more times. Work out the probability that, of the three balls Pete takes, exactly two are the same colour. (Hint – what is the alternative to 2 being the same colour?) ‘2 the same colour’ means 1 is a different colour The only other option is all 3 are the same colour So P(2 the same colour) = 1 – P(all the same colour) P(all the same colour) = P(BBB) + P(GGG) So P(2 the same colour) =

Expectation n = 200 and p = ¾ n = 306 and p = 4/5 Expectation is the long-run average you would get if a test was repeated many times If an event has probability p, the expectation in n trials is np Expectation is used as an estimate for how many times an event will occur Eg a coin is biased so that the probability of throwing heads is ¾. Dave is going to throw the coin 200 times. Work out an estimate for the number of times the coin lands on heads. n = 200 and p = ¾ so expectation = np = 200 x ¾ = 150 Eg There are 306 MPs in the Conservative Party. 4/5 of them say they support proposals to increase tuition fees. Work out an estimate for the number who will vote in favour of the changes n = 306 and p = 4/5 so expectation = np = 306 x 4/5 = 244.8 = 245 to nearest integer

Expectation 1. A coin is biased so that the probability of getting heads is 3/5. Dave is going to throw the coin 120 times. Work out an estimate for the number of times the coin lands on tails. Expectation = np = 120 x 2/5 = 48 2. The chance of rain each day in April is 2/3. Estimate the number of days you can expect rain in April. 30 days in April Expectation = np = 30 x 2/3 = 20 3. A door-to-door salesman achieves sales with a probability of 3/10. How many doors must he approach in order to expect an average of 15 sales a day? If 3/10 x n = 15 then n = = 50 doors

Expectated winnings Keith designs a game. It costs £1.60 to play the game. The probability of winning the game is 2/5 The prize for each win is £3 80 people play the game. Work out an estimate of the profit that Keith should expect to make. Takings = 80 x 1.6 = £128 Profit = takings - costs Expected winners = 2/5 x 80 = 32 Expectation = np Expected payout = 32 x 3 = £96 Estimated profit = 128 - 96 = £32

4. A fruit machine costs £1 to play and pays out £40 with a probability of 1/20. Is the machine worth playing? Explain your answer. Expected winnings each game = £40 x 1/20 = £2 But cost of game is only £1, so you can expect to win money in the long run 5. John and Tom play darts and pool every Saturday. John wins at darts 2/5 of the time and wins at pool ¾ of the time. a) Find the probability they win one of the games each. Estimate the number of times they win one of the games each, over a 60 week period Winner at darts Winner at pool a) John John b) Expectation = np Tom John = 33 times Tom Tom

Experimental probability Sometimes the probability of an event occuring is not understood (eg trying to predict the stock market!) very well. Experimental data can be collected to give an estimate of the actual probability. If an event occurs x times in n trials, the probability of the event is approximated by x/n Eg Bob is convinced his toast always lands butter side down when he drops it. He drops a piece of toast and it lands butter side down 30 times in 50 attempts. Comment on Bob’s claim. x = 30 and n =50 Bob’s claim is supported by the data, although he has not conducted that many trials so it is possible he was just unlucky. so probability The more trials, the more likely it is that the experimental data matches the actual theory Eg Bob repeats the experiment, dropping the piece of toast 1000 times. It lands butter side down 600 times. Comment on Bob’s claim now. x = 600 and n =1000 Bob’s claim is more strongly supported by the data, as it very unlikely he would be ‘unlucky’ that many times. so probability

Problem solving Eg A bag contains some red counters and blue counters. There are n red counters. There is 1 more blue counter than red counters. Bob will take a counter at random from the bag, record the colour and pick another. The probability that Bob picks two red counters is 1/6. Prove that red + blue = total P(1st pick red) = P(2nd pick red) = So P(both red) = But P(both red) , giving

Problem solving 1. A bag contains some black counters and white counters. There are n black counters. There are 2 less white counters than black counters. Bob will take a counter at random from the bag, record the colour and replaces it before picking another. The probability that Bob picks one of each counter is 3/8. Find how many of each coloured counter there are in the bag But P(one each) red + blue = total P(BW) = P(WB) = So P(one each) as n > 0 So 3 black, 1 white counter in bag

2. Gary plays two games of chess against Mijan. The probability that Gary will win any game against Mijan is 0.55 The probability that Gary will draw any game against Mijan is 0.3 In a game of chess, you score 1 point for a win, ½ point for a draw, 0 points for a loss. Work out the probability that after two games, Gary’s total score will be the same as Mijan’s total score. P(Gary loses) = 0.15 Total scores are the same if: Gary wins 1st game, Mijan wins 2nd Mijan wins 1st game, Gary wins 2nd Both games drawn

The types of people watching a film at a cinema are shown in the table. Two of these people are chosen at random to receive free cinema tickets. Calculate the probability that the two people are adults of the same gender. TOTAL = 50 P(Adult Male, Adult Male) + P(Adult Female, Adult Female) 21 50 20 49 14 50 13 49 x + x 6 35 13 175 + 43 175 (4 marks)

Misconceptions Discuss why each statement is incorrect If you toss a fair coin and get heads 5 times in a row, you are more likely to get tails the next time. The probability is the same each time- previous results are irrelevant In a football match, you can either win, lose or draw. So the probability of winning is 1/3. Winning may not have the same probability as losing You are less likely to win with lottery numbers 1,2,3,4,5,6 than if you pick numbers at random Every number has the same chance and so does every combination It might be biased, as you would only expect 25 heads, but it is still possible to get 40 out of 50 heads with a fair coin. If you toss a coin 50 times and get heads 40 times, the coin must be biased If you roll two dice and add the results, the probability of getting 9 is 1/11 as there are 11 possibilities (2-12) There are more ways to get some totals than others When tossing a coin, you are just as likely to get 5 heads in a row as 10 in a row- it’s just chance P(5 heads in a row) = 1/32 P(10 heads in a row) = 1/1024