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10/27/2015 Coin Experiment When we toss a coin in the air, we expect it to finish on a head or tail with equal likelihood. What to do: • Toss one coin 40 times. Record the number of heads in each trial, in a table: Toss two coins 60 times. Record the number of heads in each trial, in a table. Toss three coins 80 times. Record the number of heads in each trial, in a table. • Share your results for 1, 2, and 3 with several other students. Comment on any similarities and differences. • Pool your results and find new relative frequencies for tossing one coin, two coins, and three coins. •Be ready to share your results with the class. Questions on the homework? 1 10/27/2015 PROBABILITY CONTINUED Learning Target 3.2 Theoretical Probability Learning Target 3.3 Compound events C. THEORETICAL PROBABILITY The likelihood of obtaining the outcome 2 would be: The likelihood of obtaining the outcome 2 or 5 would be: This is a mathematical or theoretical probability and it is based on what we theoretically expect to occur. It is the chance of that event occurring in any trial of the experiment. Theoretically, a coin toss showing heads should occur 1 every 2 times thrown. 2 10/27/2015 SO… HOW DO QUANTIFY IT “A” is the event. 𝑈 is the sample space that event can occur in. So 𝑃(𝐴) is the probability that that event can occur in the sample space 𝑈. EXAMPLE • You have a bag of Jelly Beans and you randomly selected from the bag. It contains: • 15 Buttered Popcorn • 7 Coffee • 9 Boogers • 1) Determine the probability of getting the booger jellybean? • 2) What is the probability of not getting the coffee one? • 3) What is the probability of getting a booger one or a coffee one? • 4) what is the probability of getting a grape one? 3 10/27/2015 YOU TRY ON YOUR OWN. STOP TALKING. GRACE. • An ordinary 8 sided die is rolled once. Determine the chance of: A) Getting a 6 B) 𝑃 𝑛𝑜𝑡 6 C) Getting a 1 or 2 D) Not getting a 1 or 2. COMPLEMENTARY EVENTS • Two events are complementary if exactly one of the events must occur. If A is an event, then A’ is the complementary event of A, or ‘not A’. • 𝑷(𝐴) + 𝑷(𝐴′) = 1 • This reads, the probability of A occurring plus the probability of A not occurring equals 1. • We can solve for each of the probabilities. • 𝑃 𝐴 = 1 − 𝑃(𝐴′ ) • 𝑃 𝐴′ = 1 − 𝑃 𝐴 • Example: Emma is going to the park. She has seen her friends at the park hanging out and rollerblading 23% of the time. What is the probability that Emma will not see her friends? 4 10/27/2015 ANOTHER EXAMPLE! • Use a 2D grid to illustrate the sample space for tossing a coin and rolling a 6 sided die simultaneously. From this grid, determine the probability of: a) Tossing a head b) Getting a Tail and a 5 c) Not Getting a tail or a 6. EXAMPLE 3 • Draw a table of outcomes to display the possible results when two dice are rolled and the scores are added together. • Determine the probability that the sum of the dice is 7. • Find the probability that the dice will not add to be 8. 6 7 8 9 10 11 12 5 6 7 8 9 10 11 4 5 6 7 8 9 10 3 4 5 6 7 8 9 2 3 4 5 6 7 8 1 2 3 4 5 6 7 1 2 3 4 5 6 5 10/27/2015 D. COMPOUND EVENTS • Consider the following problem: • Box X contains 2 blue and 2 green balls. Box Y contains 1 white and 3 red balls. A ball is randomly selected from each of the boxes. Determine the probability of getting “a blue ball from X and a red ball from Y” • What can we say about these two events happening? What do we do to interpret the 2-D grid? W R R R B B G G D. INDEPENDENT EVENTS • • We can see that the area highlighted is the possible outcomes and the total area is the universal set. 6 16 • We can do this without drawing a 2D grid because they are independent compound events. • two events for which the occurrence of each one does not affect the occurrence of the other, we call these independent events. • 𝑃 𝐵 𝑎𝑛𝑑 𝑅 = 𝑃 𝐵 ∙ 𝑃 𝑅 . W • 1 2 3 4 = 3 8 R R R B B G G 6 10/27/2015 D. INDEPENDENT EVENTS EXAMPLE • Franco went to the store to buy some vegetables for his pet rabbit. Before he 1 went he found out that carrots were dirty and may cause his rabbit to get 10 2 sick. He also discovered that lettuce heads are dirty as well. Assuming that 9 he gets equal amounts of each vegetable; A) What are the chances that Franco will pull a dirty carrot and dirty lettuce? B) What are the chances of Franco not pulling a dirty vegetable? D. DEPENDENT EVENTS • A dependent event happens when one even occurs that affects the following event. • If event “A” effects event “B”, we would write: • 𝑃 𝐴 𝑡ℎ𝑒𝑛 𝐵 = 𝑃 𝐴 ∙ 𝑃 𝐵, 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 𝐴 ℎ𝑎𝑠 𝑜𝑐𝑐𝑢𝑟𝑟𝑒𝑑 • Example: • Suppose an urn contains 5 red and 3 blue tickets. One ticket is randomly chosen, its colour is noted, and it is then put aside. A second ticket is then randomly selected. • If the first ticket was red, what would the probability the second is red? • If the first ticket was blue, what would the probability be second is red? What is the probability that the first one was red and the second one is blue? 5 8 3 15 = 7 56 7 10/27/2015 E. USING A TREE DIAGRAM • Kareem is not having much luck with his motor vehicles. His car will only start 80% of the time and his motorcycle will only start 60% of the time. • Are these dependent or independent events? • Construct a tree diagram to illustrate his situation. • Use the diagram to determine the chance that • Both will start. • Kareem can only use his car. First start MC start Car M M’ Car’ M M’ ADDING EVENTS • If there is more than one outcome in an event then we need to add the probabilities of these outcomes. Example • Two boxes each contain 6 petunia plants that are not yet flowering. Box A contains 2 plants that will have purple flowers and 4 plants that will have white flowers. Box B contains 5 plants that will have purple flowers and 1 plant that will have white flowers. A box is selected by tossing a coin, and one plant is removed at random from it. Determine the probability that it will have purple flowers using a tree diagram. • We are trying to find 𝑃(𝑝𝑢𝑟𝑝𝑙𝑒 𝑓𝑙𝑜𝑤𝑒𝑟𝑠), therefore, this equals 𝑃(𝐴 𝑎𝑛𝑑 𝑃) + 𝑃(𝐵 𝑎𝑛𝑑 𝑃) • You complete the exercise. 8 10/27/2015 HW: 9C-E • C.1 #2-4 • C.2 #2 • C.3 #1,3 • D.1 #1,2,6 • D.2 #1,2,5 • E #1,2,5-7 9