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ACC Module #2 Unit 2.7

ACC Module #2 Unit 2.7

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ACC Module #2 Unit 2.7

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  1. DemingEarly College High SchoolUnit 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7 Probability Sets

  2. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7 Probability Sets Given a set of possible outcomes X, a probability distribution on X is a function that assigns a probability to each possible outcome. If the outcomes are , and the probability distribution is p, then the following rules are applied.for any i.In other words, the probability of a given outcome must be between zero and 1, while the sum of all the possible probabilities must be 1.

  3. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7 Probability Sets If is constant (or never changes), this is called a uniform probability distribution, and .For example, on a six-sided die, the probability of each of the six outcomes will be .What is the probability of a four as the outcome?What about a five?

  4. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7 Probability Sets If seeking the probability of an outcome occurring in some specific range A of possible outcomes, written P(A), add up the probabilities for each outcome in that range.For example, consider that six-sided die again, and figure the probability of getting a 3 or lower on a roll.The possible rolls are 1, 2, 3, 4, 5, and 6.. So, to get a 3 or lower, a roll of 1, 2, or 3 must be completed. The probabilities of each of these numbers being rolled is , so add these to get:

  5. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7 Probability Sets Using our honest six-sided die, what is the probability that the out come will be even? Using the same six-sided die, what is the probability that the out come will be a 1 or 6?

  6. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7 Probability Sets Using our honest six-sided die, what is the probability that the out come will be even? Using the same six-sided die, what is the probability that the out come will be a 1 or 6?

  7. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7 Probability Sets You pick a card at random.What is P(5)? 3 5 4 You pick a card at random.What is P(not a divisor of 30)? 1 2 3

  8. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7 Probability Sets You pick a card at random.What is P(5)? 3 5 4 You pick a card at random.What is P(not a divisor of 30)? 1 2 3

  9. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7 Probability Sets You flip a coin.What is P(heads or tails)? You pick a card at random.What is P(greater than 4 or prime)? 3 4 5

  10. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7 Probability Sets You flip a coin.What is P(heads or tails)? You pick a card at random.What is P(greater than 4 or prime)? 3 4 5

  11. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.1 Conditional Probabilities An outcome occasionally lies within some range of possibilities of B, and the probability that the outcomes also lie within some set of possibilities A needs to be solved. This is called a conditional probability. It is written as P(A|B), which is read “the probability of A given B” The general formula for computing condition probabilities is: Where means “A intersect B”, and consists of all the outcomes that lie in bothA and B.

  12. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.1 Conditional Probabilities However, with uniform probability distributions, simplify this a bit. Write |A| to indicate the number of outcomes in A. Then for uniform probability distributions, write:This means that all possible outcomes do not need to be known. To see why this formula works, suppose that the set of outcomes X is . Then, for a uniform probability distribution:

  13. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.1 Conditional Probabilities For example, suppose a die is rolled and it known it will land between 1 and 4. However, how many sides the die has is unknown. Figure the probability that the die is rolled higher than 2. To figure this, P(3) or P(4) does NOT need to be determined, or any other probabilities, since it is known that a fair die has a uniform probability distribution. Therefore, apply the formula . So in this case B is (1, 2, 3, 4) and is (3, 4). Therefore:

  14. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.1 Conditional Probabilities Conditional probability is an important concept because, in many situations, the likelihood of one outcome can differ radically depending on how something else comes.The probability of passing a test given that one has studied all of the material is generally much higher than the probability of passing a test given that one has not studied at all. The probability of a person having heart trouble is much lower if that person exercises regularly. The probability that a college student will graduate is higher when his or her SAT scores are higher, and so on. For this reason, there are many people who are interested in conditional probabilities.

  15. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.1 Conditional Probabilities Note that in some practical situations, changing the order of the conditional probabilities can make the outcome very difficult. For example, the probability that a person with heart trouble has exercised regularly is quite different than the probability that a person who exercises regularly will have heart trouble.Also, the probability of a person receiving a military-only award, given that he or she is or was a soldier, is generally not very high, but the probability that a person being or having been a soldier, given that he or she received a military-only award is 1.

  16. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.1 Conditional Probabilities However, in some cases, the outcomes do not influence one another this way. If the probability of A is the same regardless of whether B is given; that is, if P(A|B) = P(A), then A and B are considered independent. In this case:In fact, if it can be determined that B is also independent of A.

  17. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.1 Conditional Probabilities An example of something being independent can be seen in rolling dice. In this case, consider a red die and a green die. It is expected that when the dice are rolled, the outcome of the green die should not be dependent in any way on the outcome of the red die. Or to take another example, if the same die is rolled repeatedly, the next number rolled should not depend on which numbers have been rolled previously. Similarly, if a coin is flipped, then the next flip’s outcome does not depend on the outcomes of previous flips.

  18. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.1 Conditional Probabilities This sometimes can be counter-intuitive, since when rolling a die or flipping a coin, there can be a streak of surprising results. If, however, it is known that the die or coin is fair, then the results are just the result of the fact that over long periods of time, it is very likely that some unlikely streaks of outcomes will occur. Therefore, avoid making the thinking that when considering a series of independent outcomes, a particular outcome is “due to happen” simply because a surprising series of outcomes has already been seen.

  19. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.1 Conditional Probabilities There is a second type of common mistakes that people tend to make when reasoning about statistical outcomes: the idea that when something of low probability happens, this is surprising. It would be surprising that something with a low probability happened after just one attempt. However, with so much happening all at once, it is easy to see at least something happen in a way that seems to have a very low probability. In fact, a lottery is a good example. The odds of winning a lottery are very small, but the odds that somebody wins the lottery each week are actually fairly high. Therefore, no one should be surprised when some low probability things happen.

  20. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.2 Probability Addition Rule The addition rule for probability states that the probability of AorB happening is . Note that the subtraction of must be performed, or else it would result in double counting any outcomes that lie in both A and in B. For example, suppose that a 20-sided die is being rolled. Fred bets that the outcome will be greater than 10, while Helen bets that it will be greater than 4 but less than 15. What is the probability that at least one of them is correct?

  21. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.2 Probability Addition Rule We apply the rule: A is that outcome x is in the range x > 10, and B is that outcome x is in the range 4 < x < 15. andcan be computed by noting means the outcome x is in the range 10 < x < 15, so .

  22. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.2 Probability Addition Rule Therefore: Therefore:

  23. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.2 Probability Addition Rule The bottom line rule you want to remember: If you are asked to find the probability of event P(A)ORP(B)occurring you just add the individual probabilities. 3 4 5 You pick a card at random. What is P (greater than 4 or odd)? P(greater than four) = , P(odd) = So the P (greater than 4 or odd)?

  24. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.2 Probability Addition Rule You flip a coin. What is P(heads or tails)?

  25. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.2 Probability Addition Rule You flip a coin. What is P(heads or tails)? P(heads) = ; P(tails) = P(heads or tails)? = + = 1

  26. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.3 Probability Multiplication Rule The multiplication rule states the probability of AandB both happening is: . As an example, suppose that when Jamie wears black pants, there is a ½ probability that she wears a black shirt as well, and that she wears black pants ¾ of the time. What is the probability that she is wearing both a black shirt and black pants? Let P(A) = wearing black pants, and P(B) wearing a black shirt. We are given P(A) = ¾ and P(B|A) = ½. Multiplying the two, the probability that she is wearing both a black shirt and black pants? P(A)P(B|A) = ¾ x ½ =

  27. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.3 Probability Multiplication Rule The bottom line rule you want to remember: If you are asked to find the probability of event P(A)ANDP(B)occurring you just multiply the individual probabilities. Two events are independent if the outcome of the first event does not affect the outcome of the second event. You spin the spinner twice. What is the probability of landing on a number less than 6 and then landing on a number greater than 6?

  28. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.3 Probability Multiplication Rule The events (less than 6) and (greater than 6) are independent. The first spin does not affect the second spin. Find P(less than 6). The spinner has 8 equal sections, numbered 1, 2, 3, 4, 5, 6, 7, and 8. The numbers less than 6 are 5, 4, 3, 2, and 1. There are 5 numbers less than 6. So P(less than 6) = 5/8. Find P(greater than 6). The numbers greater than 6 are 7 and 8. There are 2 numbers greater than 6. So P(less than 6) = 2/8. The probability that the numbers are (less than 6) and (great than 6) = 5/8 x 2/8 = 10/64

  29. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.3 Probability Multiplication Rule You roll a 6-sided die two times. The die is fair and all numbers are equally probable. What is P(even, 1)? The ordered pair means P (even and 1). What is P(2, greater than 2)?

  30. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.3 Probability Multiplication Rule You roll a 6-sided die two times. The die is fair and all numbers are equally probable. What is P(even, 1)? The ordered pair means P (even and 1). What is P(even, 1)? P(even) = 3/6 = 1/2 ; P(1) = 1/6. P(even, 1) = 1/2 * 1/6 = 1/12 What is P(2, greater than 2)? P(2) = 1/6 ; P(greater than 2) = 4/6. P(2,odd) = 1/6 * 4/6 = 1/9

  31. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.3 Probability Multiplication Rule You spin the spinner twice. What is P(greater than 5, 4)? What is P(odd,6)? What is P(<2,6)?

  32. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.3 Probability Multiplication Rule You spin the spinner twice. What is P(greater than 5, 4)? P( >5 ) = 3/8; P( 4 ) = 1/8. P( >5 ) and P(4) = 3/8 * 1/8 = 3/64. What is P(odd,6)? P(odd) = 4/8 P(6) = 1/8. So P(odd,6) = 1/2 * 1/8 = 1/16. What is P(<2,6)? P(<2) = 1/8 P(6) = 1/8. So P(<2,6) = 1/8 * 1/8 = 1/64.

  33. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables The last item we will cover in probabilities, is to find “and” and “or” probabilities using two-way frequency tables. If you are asked to find the probability of event P(A)ORP(B)occurring you just add the individual probabilities. If you are asked to find the probability of event P(A)ANDP(B)occurring you just multiply the individual probabilities. You will most likely be asked to solve these type problems using a two-way frequency table. The most important thing to determine is if the are asking about event A or B, or about A and B.

  34. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables A uniform probability model is a sample space in which all outcomes are equally likely. The probability of an event A is the number of outcomes in that event divided by the total number of outcomes in the sample space. A two-way frequency table is usually depicted in a multi-layered grid like:

  35. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables A dentist was making note of her upcoming appointments with different aged patients and the reason for their visits. What is the probability that a randomly selected appointment is for a broken tooth AND is with a patient under 18 years old?

  36. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables A dentist was making note of her upcoming appointments with different aged patients and the reason for their visits. What is the probability that a randomly selected appointment is for a broken tooth AND is with a patient under 18 years old?

  37. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables Dina, an aspiring meteorologist, spent the past few weekends studying the clouds. She took detailed notes on the types of clouds observed and the time of day they were observed. What is the probability that a randomly selected cloud was marked as stratocumulus AND was observed in the morning?

  38. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables Dina, an aspiring meteorologist, spent the past few weekends studying the clouds. She took detailed notes on the types of clouds observed and the time of day they were observed. What is the probability that a randomly selected cloud was marked as stratocumulus AND was observed in the morning?

  39. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables Colin counted the boxes of cereal in a grocery store with different sizes and numbers of prizes. What is the probability that a randomly selected box of cereal does not contain one prize AND is not regular size?

  40. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables Colin counted the boxes of cereal in a grocery store with different sizes and numbers of prizes. What is the probability that a randomly selected box of cereal does not contain one prize AND is not regular size?

  41. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables Every summer, Frank goes on a week-long canoe trip with his childhood camp friends. After the trip, he usually purchases some oars from the destination city. His collection of oars is organized by year and type of wood. What is the probability that a randomly selected oar was purchased in 2012 OR was made from ash wood?

  42. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables Every summer, Frank goes on a week-long canoe trip with his childhood camp friends. After the trip, he usually purchases some oars from the destination city. His collection of oars is organized by year and type of wood. What is the probability that a randomly selected oar was purchased in 2012 OR was made from ash wood?

  43. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables Two hundred and forty students were surveyed as to their preference for a sports car or a sport utility vehicle (SUV) prior to purchasing. What is the probability that a randomly selected vehicle was purchased by a Female ORwas a Sports Car?

  44. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables Two hundred and forty students were surveyed as to which vehicle they were going to buy, a sports car or a sport utility vehicle (SUV). What is the probability that a randomly selected vehicle was purchased by a Female ORwas a Sports Car?

  45. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables Mr. Simpson conducted a survey to find out whether students would prefer to visit The Grand Canyon, Key West, Mount Rushmore, or Disney World on spring break next year. What is the probability that the most popular choice was made by a Freshman ORwas Key West?

  46. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.7.4 Using Two-Way Frequency Tables Mr. Simpson conducted a survey to find out whether students would prefer to visit The Grand Canyon, Key West, Mount Rushmore, or Disney World on spring break next year. What is the probability that the most popular choice was made by a Freshman ORwas Key West?

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