Don ’t leave it to chance!

1. Don’t leave it to chance!

2. What is probability?

3. Simple Questions • If I flip a coin, what is the probability of getting heads? • What is the probability of picking a red marble out of 5 red marbles and 20 blue marbles?

4. Answers • A coin has two sides (head, tail) – Probability of head = • Probability of red marble = =

5. Probability • Likelihood (chance) of something to occur • Ratios, fractions, decimals, percentage • ¼ = 1:4 = 0.25 = 25%

6. Applications • To make predictions based on likelihood Eg) The chance of Manchester United beating Liverpool • To help in the planning of future events Eg) Planning outdoor activities based on the probability of raining, etc. OR

7. Is this mere chance? • You’re on a game show. You are given a choice of three doors, 2 with a goat behind and 1 with a car. You get whatever is behind the door. You pick a door and the host then reveals another door with a goat. Does it make any difference if you switch? If yes, should you? (Of course you want the car, not the goat.) OR

8. Well… • For those of you who think that no difference is made, you better learn your probabilities!  • http://www.youtube.com/watch?NR=1&feature=endscreen&v=mhlc7peGlGg

9. Terminology

10. Terminology • Event: A possible outcome • Independent events: describing 2 or more events not affecting each other (for example, probability of getting 2 on a dice and probability of getting heads on a coin)

11. Terminology • Mutually exclusive events: Events that cannot happen at the same time • P(A): Chance of A happening • P(A’): Chance of A not happening. Thus, P(A’)+P(A)=1

12. Basic Concepts

13. Basic concept- 1 • The maximum probability of an event happening is 1. (100%) • As such, addition of the probability of all possible events should add up to 1. • Consequently, P(A)+P(A’)=1, because A and A’ includes all possible events. A’ A

14. Basic concepts- Randomness • Probability has an element of randomness • This means that the events can be predicted by probability but it may not be correct all the time

15. Basic concepts- Addition of probabilities • Probability that two or more events will occur (A will occur or B will occur) • P(1 or 2 appearing on a 6-sided dice)= 1/3. • P(1 appearing on a 6-sided dice)=1/6 • P(2 appearing on a 6-sided dice)=1/6 • 1/6+1/6=1/3

16. Basic concepts- Addition of probabilities • The events must be mutually exclusive (i.e. they cannot be something that can happen together) • Thus the probability that A occurs or B occurs is given by: • P(A or B)=P(A)+P(B)

17. Basic concepts- Multiplying probabilities • We multiply probabilities to find out the probabilities of two or more independent events happening together

18. Basic concepts- Multiplying probabilities • Eg.) There is a 50%, 30% and 70% that it will rain on Monday, Tuesday, Wednesday respectively. What is the chance that it will rain on all days? • 50%x30%x70%=105/1000=10.5% OR

19. Basic concepts- Multiplying probabilities • Hence the probability of event A and B both occurring is given by • P(A and B) = P(A) x P(B), where A and B are two independent events • We add probabilities when we find P(A or B) happening; we multiply them when we find P(A and B) happening.

20. Ways of finding Probabilities

21. Ways of finding probabilities • Probability Diagram • Tree Diagram • Solve using identities

22. Ways of finding probabilities • Probability diagram: Show all possible events of an experiment • Two fair 6-sided dices are thrown together. Find (a) The probability that the addition of the two values obtained is even (b) The probability that the addition of the two values obtained is prime

23. Probability diagram First dice (a) We get (even results)/ All events =18/36=1/2 (b) (prime results)/ All events =15/36=5/12 Bold- Even Underlined- Prime Second dice

24. Ways of finding probabilities • Tree diagram: See all possible events • Each branch represents a possible outcome • The following graph shows a diagram of a coin being tossed three times:

25. Tree diagram

26. Tree diagram • From the tree diagram, we can see eight possible outcomes. To find out the probability of a particular event, we need to look at all the events. • The sum of the probabilities for any set of events is always 1.

27. Worksheet Time!

28. Real-life examples • What is the probability of getting a straight flush after randomly drawing cards from a standard deck of 52 cards? • What is the probability of getting a straight flush after randomly drawing cards from two standard decks of 52 cards?

29. Real-life examples • What is the probability of a woman who had 3 baby boys giving birth to a girl next? • You are betting with your friend on the sum of the numbers shown after throwing two 6-sided dices. Does every number stand an equal chance? Explain. If your answer is no, why? OR

30. Thank you!