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Probability

Probability. What is ... Probability? a measure of how likely it is that an event will happen . Experiment? something like rolling a die, drawing a card, tossing a coin. A trial? a single part of an experiment which consists of many trials. ie . rolling a die 6 times is 6 trials.

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Probability

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  1. Probability

  2. What is ... Probability? a measure of how likely it is that an event will happen. Experiment? something like rolling a die, drawing a card, tossing a coin. A trial? a single part of an experiment which consists of many trials. ie. rolling a die 6 times is 6 trials. An outcome ? the result of a trial. i.e. rolling a 5. sample space (S)? the set of all possible outcomes of an experiment. i.e. S = {(1), (2), (3), (4), (5), (6)} event (E)? is part of the sample space. i.e. E = {Throwing an even number} = {(2), (4), (6)} Definitions

  3. Experimental Probability At the fair I am playing a lucky dip game where I must throw a square card into a sandpit. If I land on a number, I win a prize. If I throw 20 times and land on a number 3 times, what is the probability that if I throw again I will a. win a prize? b. Not win a prize? • 3/20 • 17/20

  4. Try these • 230 out of the 250 people manage to get at to the airport one hour before their flight • What is the probability that a person chosen at random gets to the airport 1 hour before their flight? • What is the probability they don’t?

  5. Equally Likely Outcomes The red cards of a pack of cards are placed in a bag. What is the probability that when you draw a card out, • you draw an picture card? • A diamond? • An even diamond?

  6. Expected Number of Outcomes If a die is tossed 25 times how many 3’s would we expect to get? Expected number of 3’s = 1/6 25 = 4.167 = 4 times(0dp). If spin a spinner with 4 equal sectors, how many red’s would you get in 12 spins? Expected number of reds = 1/4  12 = 3 times.

  7. Example A mouse in a maze. It gets to an intersection where it can turn left or right. It makes a turn and then comes to another intersection where it can turn left or right again. It makes another turn. Draw a probability tree to show the probability that the mouse turned: a) right and then left in that order. b) Right and left in any order. Answer • (¼) • (¼ + ¼ = ½)

  8. With replacement. In Ms Hutton’s drawer there are 5 blue, 4 white and 6 red scarves. She pulls 1 scarf, note it’s colour, replaces it and then pull out another scarf out of the bag. Draw a probability tree to show the different outcomes and hence determine the: • P(2 blue) = 5/15 5/15 = 1/9 • P(2 white) = 4/10 4/10 = 4/25 • P(2 red) = 3/5 3/5 = 9/25 • P(1 red and 1 white in any order) = P(RW) + P(WR) = 6/15 4/15 + 4/15 6/15 = 16/75 • P(1 yellow and 1 red in any order) = 0

  9. Try this • 0.09 • 0.49 • 0.595 On the ski trip they buy fish and chips or sausages and chips for the tea on the way to Queenstown. 60% want fish for their first course and the rest want sausages. 85% of them have an ice-cream sundae for their second course and the rest have a pineapple fritter. What they have for their second course does not depend on what they have for their first course. Some of the information is shown on the diagram below What is the probability that a person has: • fish and chips and a pineapple fritter? • either sausage and chips or a pineapple fritter? The probability that a person has a coke is 0.7 What is the probability that if a person has fish and chips, they will also have an ice cream sundae and a coke?

  10. Conditional probabilities This tree diagram shows the probability of a student buying Fish or Chips before the Year 11 dance. We are told that the student does not take Fish, so we only look at these students. P (buys Chips, given doesn’t buy Fish) = 7/12 What is the probability that a student who buys Fish, also buys Chips? Student buys Fish so only look at them. P (Student buys Fish and Chips also) = 6/8

  11. Without replacement. In Ms Hutton’s drawer there are 5 blue, 4 white and 6 red scarves. She pulls 1 scarf, does not replace it and then pull out another scarf out of the bag. Draw a probability tree to show the different outcomes and hence determine the: • P(2 blue) = 2/21 • P(2 red) = 1/7 • P(2 yellow) = 0 • P(1 blue and 1 yellow in that order) = 2/21 • P(1 blue and 1 yellow in any order) = 8/45

  12. Try these A box of chocolates contains 9 hard centres (H) and 6 soft centres (S). One chocolate is taken at random and eaten, then a second chocolate is taken. Find the probability that: • Both chocolates have soft centres • One has a hard centre and one has a soft centre

  13. HoytsPART ONE The local Hoyts 6 is planning a promotional give‑away to mark the opening of the movie Lord of the Rings. Every adult movie ticket purchased in the month of October will be accompanied by one of six figurines based on characters from the movie Gandalf, Aragorn, Boromir, Legolas, Gimli, and Frodo. The allocation of the movie figurines is done on a completely random basis. You want to collect the whole set of 6 figurines. Investigate how many movie tickets you would need to purchase in order to complete your collection.

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