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Probability

Probability. If you roll a 2 you will get no homework. What are your chances of getting no homework?. First we need to find out what could happen when we roll the dice. Find all of the possible things that could happen . These are called the outcomes.

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Probability

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  1. Probability If you roll a 2 you will get no homework. What are your chances of getting no homework? First we need to find out what could happen when we roll the dice. Find all of the possible things that could happen . These are called the outcomes. The specific outcome we are looking for is 2. This is called the event. In this experiment we have 1 chance of getting a 2 The outcomes are1,1:1,2:1,3:1,4:1,5:1,6 If we continue this pattern we will have 36 possible outcomes.

  2. In words, the probability of an event is the ratio of the number of ways an event can occur to the number of possible outcomes. The number of ways the event can occur In symbols P(event) = Number of possible outcomes In our experiment So the probability of getting no homework is 1 P(2) = 36 one in thirty-six Course 1 wb 14-1

  3. Sample Space and Outcomes Drinks Main Dish Desert The cafeteria is offering the above items for lunch. For $ 3 you get one item from each section. If the menu stays the same how many different meals could you eat? One way to find out is by listing all of the possible outcomes which is called the Sample space.

  4. Tree Diagrams By following down the branches you can determine all of the possible choices or outcomes and see the entire sample space. Course 3 wb 13-1

  5. A tree diagram can be made horizontally and by using words or letters to represent the choices. Drinks Main Dish Desert Outcomes HB M HB,PIE,M PIE HB,PIE,IT IT HB P HB,IC,M M IC IT HB,IC,IT HD M P,PIE,M PIE PIE IT P P,PIE,IT P,IC,M M IC IT P,IC,IT IC M HD,PIE,M PIE IT M HD,PIE,IT HD M IC HD,IC,M IT Course 2 wb13-1 IT HD,IC,IT

  6. Fundamental Counting Principle When looking for just the total number of outcomes, using the Fundamental Counting Principle is quick and easy. Main dishes Drinks Deserts 3 x 2 x = 12 2 The lunch problem had 3 groups of choices. Multiplying the number of choices in each group provides us with the number of outcomes. There are 12 different lunch combinations. Course 2 wb13-2 course 3 13-1

  7. Theoretical Probability Theoretical probability is a mathematical computation using the ratio The number of ways the event can occur Number of possible outcomes This type of probability can allow us to predict what could happen. But what actually happens may be different. If a coin is flipped we can get heads or tails. Our theoretical probability of heads is 1/2.To find out how many heads we could expect in ten trials (flips) multiply 1/2 by 10 (5 heads).

  8. Multi-Stage Experiments Roll the die Flip the coin Spin the spinner To find P(1,3,H) use the fundamental counting principle P(1) P(3) P(H) 1/6 x 1/4 x 1/2 P(1,3,H) = 1/48 This is the theoretical probability. Course 3 13-5 course 2 13-6

  9. Independent and Dependent Events The previous experiment is an example of an independent event. (The roll of the die had no impact on the the spinner or the coin) Experiment: Pick a ball from box 1 and place it in box 2. Then pick a ball from box 2. What is P(B)? In box 1 a black or white ball can be picked.What happens in box 2 is dependent on what happens in box 1 since the # of each color will change depending on what is picked from box 1. This experiment is a Dependent Event.

  10. A tree diagram can be very helpful in working with Dependent Events. First, draw the diagram and list the P on each branch. Next label the possible outcomes. OUTCOMES P W WW 2/3 x 2/4 = 4/12 2/4 2/3 W 2/3 x 2/4 = 4 /12 WB B 2/4 1/4 1/3 x 1/4 = 1/12 BW 1/3 W B 1/3 x 3/4 = 3/12 BB B 3/4 Then multiply along the branches to get the P of each outcome. Finally, add the events that end in B:4/12 +3/12 =7/12

  11. Experimental Probability Theoretical probability is based on mathematical principles. Experimental Probability is an estimated probability based on the relative frequency of positive outcomes occurring during an experiment. Experimental probability does not always coincide with theoretical probability. Many organizations use experimental probability to make predictions or forecasts of future trends. Surveys are often used to obtain the data for the basis of the experimental probability. Course 2 wb13-3

  12. Below are the results of a survey of the 6th and 7th grade students in Norwood. If there are 325 - 6th graders and 350 - 7th graders in the Valley, about how many of each class will prefer KROC? KROC 92.3 WPLJ 95.5 Z 100 KTU 103.5 Q 104.3 6TH GRADERS 24 30 13 43 21 7TH GRADERS 26 32 15 38 19 The survey represents a sample population. We can use this data to obtain an experimental probability. However, it is only useful when applied to a similar population. (For example using this data to predict what station their parents might prefer would not be useful because the population “parents” is not similar to the population “6th and 7th graders”)

  13. KROC 92.3 WPLJ 95.5 Z 100 KTU 103.5 Q 104.3 6TH GRADERS 24 30 13 43 21 7TH GRADERS 26 32 15 38 19 First, get a total of each population 6th = 131 7th = 130( these will be our denominators) Next, find the event we are trying to find (KROC92.3 6th = 24 7th = 26). These will serve as our numerators. 24 26 131 130 Then multiply the ratios by the total number of the new population. 24 x 325 = 60 26 x 350 = 70 131 130 Course 3 wb 13-7 course 2wb13-5

  14. Probability and Area A sky diver is going to land in the school field. If it is equally likely to land on any part of the field what is the probability he will land in a circle in the middle of the field? 400 feet Area of circle Area of rectangle P(circle) = 125 feet 42 feet A = Pi r * r =3.14 *21 * 21 =1385 sq ft. P(circle) = 1385 50,000 = 2.8% A = l * w = 400 *125 = 50,000 sq ft Course 1 wb14-4

  15. Permutations Belgium The flags of Germany and Italy are made of black, yellow, red, green and white stripes stripes. Italy How many different flags can be made of these colors using only vertical stripes? For the 1st color we have 5 choices. After the 1st stripe is used we have 4 choices for the next stripe. Then we have 3 colors left for the 3rd stripe. To solve we can use the counting principle 5 x 4 x 3 = 60 different flags.

  16. This problem is an example of a permutation. A permutation is an arrangement of objects in which order is important. Notice that black,yellow and red is not the same flag as red, yellow and black. In our problem we have 5 choices to be taken or used 3 at a time. The permutation formula is P(n,r) n is the number of items or choices and r is how many are used or taken at a time If we were making a flag with 4 stripes our formula would be: P(5,4) 5 colors(choices) taken 4 at a time (4 stripes) 5 x 4 x 3 x 2 = 120 different flags. Course 2 wb 13-7 course 3 wb 13-2

  17. Factorial How many 4 digit numbers can be made from 3, 5 ,7 ,9? This is a permutation of 4 things taken 4 at a time P(4,4) Another way of writing this is 4! To solve we multiply 4 x 3 x 2 x 1 This is read 4 factorial. In words n! is the product of all of the counting numbers starting with n and counting backwards to 1. 6! 4! In this problem we have 6 x 5 x 4 x3 x2 x 1 4 x 3 x 2 x 1 In this problem we can simplify. Our answer is 6 x 5 = 30

  18. Combinations Joe, Sam, Tom and Bob are all guards for the basketball team. In how many ways can the coach choose 2 starting guards? In this problem Joe and Tom are the same as Tom and Joe. So the order is not important. We ca make a list:JS, JT, JB,ST, SB, TB. These are all of the different combinations since JS and SJ are the same. So we have six different combinations.

  19. This problem is an example of a Combination. A combination is an arrangement in which the order is not important. In our problem whether we list Tom and Joe or Joe and Tom, it is still the same combination of players. The format for combinations is C(n,r) which in words means the number of combinations of n things taken r at a time. P(n,r) C(n,r) = Mathematically we write it: r! The basketball problem can be solved like this: C(4,2) = P(4,2) = 4 x 3 = 12 = 6 combinations. 2! 2 x 1 2 Course 3 wb 13-3 Course 2 wb 13-8

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