Probability

1. Probability Year 9

2. What are the chances? What is the probability of winning the first prize in Lotto? Why to people toss a coin to make a decision? What other tools do we use in a game of chance?

3. Decide whether the chance of each of the following happening is less than ½, equal to ½ or greater than ½. 1.) Toss a fair coin and it comes up heads 2.) Throw a six-sided die and a 4 will turn up. 3.) The temperature in Dunedin on Xmas day will be -10° 4.) You win a prize if you buy 80 out of a possible 100 tickets for a raffle. 5.) If you throw a 10-sided die, an even number will come up. = 1/2 < 1/2 < 1/2 >1/2 =1/2

4. Exploring Probability Construct a table in EXCEL or your notebook similar the one on the right Roll a die 6 times and record each result in Row 1 This is 1 trial. Repeat this process 10 times. That’s a total of 60 rolls! Start Rolling !

5. Exploring Probability How do your results compare with what you expected? Compare your results with another student.

6. Exploring Probability List all the possible totals you can roll with two normal dice in a table. 2 Roll two dice 50 times and record the total of the numbers on the dice in a table. 9 12 2 11 12

7. Exploring Probability 1.) From your table what is your experimentalprobability of rolling: a.) a six? b.) a one? c.) a total less than 4? d.) an even number? e.) not an even number? f.) What happens when you add your answers from part d and e ? g.) Is 1 a possible outcome?

8. Note 1: Probability…. is all about chance or how likely an event will occur. This scale shows how we can describe the probability of an event

9. Note 1: Probability…. Describe how likely these events are: a.) It will rain in Dunedin this year. b.) You will have a sandwich for lunch. c.) You wear a uniform to school. d.) You help to clean up after a meal. e.) There will be 2 full moons this month. f.) You toss a coin and it comes up heads. g.) You win the lottery. IWBEx 33.02 Pg953-954

10. Note 2: Key Ideas Probabilities can be written as fractions, decimals or percentages. Probabilities are always between 0 and 1. The sample space is a list of all possible outcomes. The probability of all possible outcomes always add to 1.

11. Probability - example e.g. A jar contains a large number of marbles coloured red, green, yellow, orange and blue. A marble was chosen at random, its colour noted and then replaced. This experiment was carried out 200 times. Here are the results. What is the probability that a randomly selected marble is: a.) orange b.) green c.) not green d.) red or blue = = = =

12. Note 2: Probability…. 50 vehicles passed Simon’s house over the past 2 hours and the results are recorded in the following table. How likely do you think the next vehicle that passes will be: a.) a car b.) a bus c.) not a motorbike likely not very likely almost certain IWBEx 33.03 Pg956-959

13. Long run Probability Task Experimental probability from an experiment repeated a large number of times can be useful to make predictions about events. Toss a coin 100 times. Record how many times it lands on heads in a table in EXCEL. After every 10 throws calculate the fraction of heads so far. Convert your proportions (fractions) to decimals. Graph the number of throws vs. the proportion of heads.

14. Your table should look like this……. Graph the number of tosses vs. the proportion of heads What do you notice about the proportion of heads tossed as the number of tosses increases?

15. Your table should look like this……. How many heads would you expect to get when you throw a coin 200 times? Compare your results with the theoretical probability of throwing a head.

16. Note 3: Long Run Probability • Long run proportions can be obtained by repeating the experiment a number of times • there will always be some variation in experiments because chance is involved • probability becomes more accurate as more trials are carried out (closer to theoretical probability) Probabilities can be written as fractions, decimals or percentages. Probabilities are always between 0 and 1. The sample space is a list of all possible outcomes. The probability of all possible outcomes always add to 1.

17. Note 3: Long Run Probability • Long run proportions can be obtained by repeating the experiment a number of times • there will always be some variation in experiments because chance is involved • probability becomes more accurate as more trials are carried out (closer to theoretical probability) IWBEx 33.04 Pg962-964

18. Note 4: Equally Likely Outcomes When outcomes of an event are equally likely, their probabilities are the same. If A is a particular event then: P(A) = P(A) means ‘the probability that A will occur’ The compliment (opposite of A) is all the possible outcomes not in A and is written A’ (not A) P(not A) = 1 – P(A) IWBEx 33.05 Pg968-971

19. Note 5: Tree Diagrams • Tree diagrams are useful for listing outcomes of experiments that have 2 or more successive events • (choices are repeated) • the first event is at the end of the first branch • the second event is at the end of the second branch etc. • the outcomes for the combined events are listed on the right-hand side.

20. Note 5: Tree Diagrams

21. Note 5: Tree Diagrams How many possible results are there? 2 x 3 = 6 R SR H SH Find(random select) P(single) = P(single vanilla)= P(raspberry)= DV V D R DR H DH =

22. Note 5: Tree Diagrams IWBEx 33.06 Pg974-977 Draw a tree diagram to show some alternative ways you could spend your Saturday You must first do a chore (choose from 4 options), Then you can choose to either watch a DVD or visit friends (2 options) 4 x 2 = 8 possible outcomes