1. Probability NCEA Level 1

2. There are 32 students in this class. 15 have seen “the Dark Knight”. 12 want to see the movie but haven’t. 5 don’t want to see the movie. If one random student is selected. What’s the probability he or she wants to watch the movie?

3. Definitions  experiments : Corne and Blake toss a coin to decide who get to go to the bathroom first Rya draws a Tarot card to predict her future  Trial One performance of the experiment: One performance of the experiment: Corne and Blake toss a coin 3 times ( 3 trials) Corne and Blake toss a coin 3 times ( 3 trials) Madison has bought lotto ticket 30 times (30 trials)  outcome The result of a trial of an experiment The result of a trial of an experiment Rya draws out a “knight” from her tarot set. Rya draws out a “knight” from her tarot set. The coin landed on head. (Blake gets to go first) The coin landed on head. (Blake gets to go first)

4.  Sample space The set of all the possible outcomes of an experiment. The set of all the possible outcomes of an experiment. Tossing a coin—Head and tail Tossing a coin—Head and tail Drawing a Tarot card –The sun, the moon, the knight, the queen, the king……a total of 25! Drawing a Tarot card –The sun, the moon, the knight, the queen, the king……a total of 25!  Event The combination of one or more outcomes The combination of one or more outcomes Rya gets both a ‘knight’ and a “prince” Rya gets both a ‘knight’ and a “prince”  Frequency The number of times the event occurs The number of times the event occurs The number of time Madison wins lotto (0) The number of time Madison wins lotto (0) The number of times the coin lands on head. The number of times the coin lands on head.

5. Probability Is expressed as fraction or percentage.  P(E) =

6. Time spent on maths (hours) Frequency 0.1-0.253 0.25-0.515 0.5-17 1-1.53 1.5-22 P( 1.5-2 hours)= P( 1.5-2 hours)= P(less than 0,5)= P(less than 0.1)= P(more than 0.1)= Ms Li’s 11 Mat class has been surveyed On how many hours they spend on maths Per day. There are 32 students in her class.

7. Real life  In many real life situations we can only estimate the probability of an event. We do this by collecting data and working out proportions.  Example:  The table shows the type of books on a bookshelf. Hard back Paper back total Non- fiction 6410 fiction22022 total82432

8. Hard back Paper back total Non- fiction 6410 fiction22022 total82432 (a)If one book is taken off the shelf at random, calculate the P(paperback, non-fiction) P(fiction)

9.  Jack has been keeping records of his wins and losses for the different kinds of games he plays, in order to find out in which games he has most success. His tabulated records are shown below: WinsLossesTotal Ball games 13070200 Card games 14158199 Board games 10075175 Total371203574

10. WinsLossesTotal Ball games 13070200 Card games 14158199 Board games 10075175 Total371203574 a.What is the probability that Jack wins a board game? b.What is the probability that the last game Jake won was a board game?

11. 16 32 8 8 8 8 8 8 8 8 16

12. Probability Tree

13.  A bag contains three red balls, four blue balls and five green balls. If two balls are drawn, with replacement, draw a probability tree to find out the probability that:  a. The first ball will be green? b. The first ball is red and the second ball is blue? c. Both balls are red? d. Both balls are not green?

14. Tree Diagrams  School is selling ‘scratch & win’ cards as a fundraising project. Each card has a chance of 1/3 of winning a prize. Hoony has bought 2 cards.  (a) Show the outcomes in a tree diagram each with their probability of happening.  (b) Calculate P(one prize)  (c) Calculate P(at least one prize)

15. Mikhail and Robert play two games of tennis. The probability of Mikhail winning the first game is 1/3. If he wins the first game, the probability of him winning the second game is 2/3, but if he loses the first game the probability of him losing the second game is 1/2. (a)Draw a probability tree for these two games (b)What is the probability of Mikhail winning both games? (c)What is the probability of Mikhail losing both games? (d)What is the probability of Robert winning one game and losing one game?

16. Sampling without replacement In this type of probability tree we DO NOT put the object back so there are fewer to select from the next time round. Each time we select a choice it is then gone and the subsequent number of options is reduced. Probabilities on following branches are changed.

17. Eg marbles in a bag 3 red, 2blue, 5 green. We take out one and then put it back and then take out another and put it back. Out comes

18. Eg marbles in a bag 3 red, 2blue, 5 green. We take out one and then DON’T put it back and then take out another.

19. Jaz goes to Rome. She has a guide book which lists 19 important statues. 9 were erected in the 19 th century and 10 were erected in the 20 th century. If she visits two statues at random what is the probability that both were erected in the 19 th century

20. YearBrownHair Black Hair OtherTotal 1998450200120 1999425250100775 200042527585 2001400200100700 2002375225125 Total1150 What is the probability that a person enrolling at Last Chance College: a) In 2001, will have brown hair? (conditional probability) b) In 2000, will not have black hair? (conditional probability) c) In the 5 year period would have had brown hair?

21. TallDwarfTotal Red232245 Blue451560 Yellow372865 10565170 A scientist plants seeds. When the seeds came up he recorded the results. h) A blue flower is dwarf (conditional) i) A yellow flower is tall (conditional) j) A flower is yellow

22. Conditional Probability

23. 2) Given that the first marble drawn out was green, what is the probability that the second marble is red? Given the first marble was green, tells us that we can miss out the first part of tree and start at the second branch. If you already know some conditions then it eliminates part of the tree so we only need to consider the relevant parts There are 6 red marbles and 4 green marbles in a bag. 1) What is the probability of drawing out a red and a green marble?

24. Students were asked if they had takeaways in the last week 45% of the students were male. if they were male the probability they had takeaways was 40%. If they were female the probability that they had takeaways was 30% If the females had takeaways there was a 25% chance they had fish and chips. If they were male there was a 35% chance they had fish and chips 1) Draw a probability tree to represent this situation

25. 2) What is the probability that a student chosen at random has fish and chips?

26. 3) Given that a student is female what is the probability that they has fish and chips? As it is given they are female we start after that branch

27. The most common campervan booking at SeeNZ is for a 10-day length of time.85% of customer requests for a 10- day booking can be confirmed immediately.The rest of the customer requests for a 10-day booking go on a waiting list. Only 25% of those on the waiting list eventually have their request for a 10-day bookingconfirmed. SeeNZ had to turn away 20 people during a three month period, because their request for a 10-day booking could not be confirmed. Calculate the total number of customer requests for a 10- day booking SeeNZ received during that three month period. 2007 Excellence Question 177

29. PROBABILITY SIMULATIONS This is when we construct an experiment to give us an idea of what might happen in a particular situation.

30.  Image if Robert and Murphy play table tennis. There is 50% and 50% chance of winning for both of them. How would you simulate the game 10 times? What tool would you use to help you? There is 50% and 50% chance of winning for both of them. How would you simulate the game 10 times? What tool would you use to help you?  One out of Six people will be randomly chosen to go on a field trip. How would you simulate the choosing process?  Every year Nick, Matt and Charlotte compete for a maths scholarship. Nick and Matt both have 25% chance of getting the scholarship. Charlotte has 50% chance of getting the scholarship. How would you simulate the situation for 6 years?

31. Key Concepts  Tools What are you using – die, cards, spinner, calculator (Ran#) What are you using – die, cards, spinner, calculator (Ran#)  Trials Define each trial Define each trial How many trials will be needed How many trials will be needed  List (Results) List the results in tables List the results in tables  Calculation Answer questions by calculating the probability Answer questions by calculating the probability

32. Simulation Question Simulation Question  There are three sets of lights along Maths Highway. Each light is red 30% of the time.  Design and carry out a simulation to predict how many cars out of 50 would go along the highway without stopping at any lights.

33. TOOL  Use a calculator to create random numbers from 1 to 10  with 1, 2, 3 representing -Stop (Red) and 4, 5, 6, 7, 8, 9, 0 representing Go (Green)

34. Trial  Select 3 random numbers and record them. Each number represent a different colour light.  If all 3 are GO put a tick in the result column. Repeat the trial 50 times.

35. Trial OUTCOME LIGHTResult 1st#2nd#3rd#1st2nd3rd 1163stopGostop 2748GoGoGo 3 301stopGostop 4 5 6 7 8 9 50 Results

36. CALCULATION  P(Not Stopping)= No of  50

37. Madison wants to go on a cruise with 6 of her friends. There is a special discount for groups of 5. She can go if she can get 4 other friends to go. There is a 50% chance that each of her friends will go. a) Find and estimate of the probability that she will have at least 4 of her friends will go b) What is the most likely number of friends that will want to go? 1)Tool: Flip a coin : Heads go, Tails not go 2) Trial: Flip a coin 6 times- one trial Perform 50 trials 3) Results123456 No. that go At least 4 go HHTTHH4Y THTHTH3N HHHHHT5Y etc

38. 4. Calculation a)no. of Y 50 b) The most frequently occurring number of friends is the most likely number that will want to go

39. Example 2  Five cards are randomly inserted into cereal packets. They each carry one of the letters P, R, I, Z or E. What is the probability that all five cards are obtained if 10 cereal boxes are bought.  A TOOL is needed to simulate the situation  A TRIAL needs to be described  LIST. How will data be recorded?  CALCULATION p(all 5 cards obtained) = no. successes/30

40. A dog breeder wants to know the average number of puppies produced by his 600 dogs over a period of three breeding seasons. Assume that each dog produces either a single puppy or twins. From the record of past breeding seasons the breeder know that the probability of having twin puppies is 1/5. Design a model to simulate the breeding of puppies over three successive breeding seasons. Use the results of your simulations to find the mean number of puppies produced by a dog over the three seasons. The breeder decided if the dog produced two sets of twins will no longer be used for breeding purposes. Use theoretical probability to find how many of the 600 dogs will no longer be suitable for breeding after three years.

41. Simulation guard and thief

42. The Guard and the Thief An office building has a cash register and a safe in two different places. The register has $2000, the safe has $10,000. Every night, there is a guard and a thief. The guard spends 2/3 of the time by the safe, 1/3 of the time at the cash register. The thief only goes to the cash register 1/6 of the time. 5/6 of the time he goes to the safe. At the end of ten days, how much money will the thief have stolen? (Assume that he doesn’t get caught in these 10 days)

43.  Tool: Use a dice.  Trail One trail is one throw of the dice. One trail is one throw of the dice. Guard: Guard: at cash register: at cash register: throw 1 and 2 throw 1 and 2 at safe: at safe: throw 3, 4, 5 and 6 throw 3, 4, 5 and 6 Thief: Thief: at cash register: at cash register: throw 1 throw 1 at safe: at safe: Throw 2,3,4,5,6 Throw 2,3,4,5,6  Tool Use calculator to generate random numbers from Use calculator to generate random numbers from 1-6. 6  Ran#+1 1-6. 6  Ran#+1  Trial One trail is generating 1 random number One trail is generating 1 random number Guard: Guard: at cash register: Generating 1 and 2 at safe: Generating 3, 4, 5 and 6 Thief: Thief: at cash register: Generating 1 at safe: Generating 2,3,4,5,6

44. Trial no. GuardTheif Money taken 1 2 3 ….. 10 Total money 2 (cash)3 (safe) 5 (safe) 4 (safe) 0 10,000 3 (safe) 1 (cash) 2000