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Probability

Probability. Would you bet your life?. Humans not only bet money when they gamble, but also bet their lives by engaging in unhealthy activities such as smoking, drinking, using drugs, and exceeding the speed limit when driving.

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Probability

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  1. Probability

  2. Would you bet your life? • Humans not only bet money when they gamble, but also bet their lives by engaging in unhealthy activities such as smoking, drinking, using drugs, and exceeding the speed limit when driving. • Many people do not care about the risks involved in these activities since they do not understand the concepts of probability. • On the other hand, people may fear activities that involve little risk to health or life because these activities have been sensationalized by the press and media. • In this session, we will learn about probability.

  3. Probability • Probability can be defined as the chance of an event occurring. • It grew out of the use of cards, coins and dice in games of chance.

  4. Probability • The study of probability helps us figure out the likelihood of something happening. • For instance, when you roll a die, you might ask how likely you are to roll a five. • In math, we call the "something happening" an "event." • The probability of the occurrence of an event can be expressed as a fraction or a decimal from 0 to 1. • Events that are unlikely will have a probability near 0, and events that are likely to happen have probabilities near 1.

  5. Notation for Probabilities • P - denotes a probability. • A, B, and E - denote specific events. • P (A)- denotes the probability of event A occurring.

  6. Probability 0.5 Impossible to occur Certain to occur Even chance • Range of Probabilities Rule • The probability of an event E is between 0 and 1, inclusive. That is 100% 0% 0  P(E)  1.

  7. Probability experiments • A process (or an activity) such as flipping a coin, rolling a die, or drawing a card from a deck is called a probability experiment (activity). • In other words, A probability experiment is a chance process that leads to well-defined results called outcomes. • The result of a single trial in a probability experiment is the outcome. • A trialmeans flipping a coin once, rolling one die once, or the like. • When a coin is tossed, there are two possible outcomes: head or tail. • ( Note, we exclude the possibility of a coin landing on its edge.) • Rolling a single die, there are six outcomes: 1 ,2 , 3 , 4, 5, 6.

  8. Probability experiments • In any experiment, the set of all possible outcomes is called the sample space. • Experiment (Activity) Sample Space • Toss one coin Head,Tail (H,T) • Roll one die 1, 2, 3, 4, 5, 6 • True/false question True, False • Toss two coins HH, TT, HT, TH

  9. Example • Find the sample space for the gender of the children if a family has three children. Use B for boy and G for girl. Third Child Outcomes BBB BBG BGB BGG GBB GBG GGB GGG First Child Second Child B B G B G B G G B B G B G G

  10. Example • Find the sample space for rolling two dice • Die 2 • Die 1 • 1 2 3 4 5 6 • 1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) • 2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) • 3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) • 4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) • 5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) • 6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

  11. Events An event is a set of outcomes of a probability experiment. we call the "something happening" an "event."An event can be one outcome or more than one outcome. • Example: A die is rolled. • Event E is rolling the number 6. • a single outcome, so we can E a simple event. • 2. Event A is rolling an even number. The outcomes of event A are {2, 4, 6}, so event A is not a simple event.

  12. Types of Probabilities • 1. Classical Probability (or Theoretical Probability) • 2.Empirical Probability ( orRelative Frequency Approximations of Probability) • 3. Subjective Probability

  13. Classical Probability • One event, all outcomes equally likely • Suppose we have a jar with 4 red marbles and 6 blue marbles, and we want to find the probability of drawing a red marble at random (E). In this case we know that all outcomes are equally likely: any individual marble has the same chance of being drawn. • To find the probability of an event E with all outcomes equally likely, we use a fraction: • What's a favorable outcome?In our example, where we want to find the probability of drawing a red marble at random, our favorable outcome is drawing a red marble. • What's the total number of possible outcomes (sample space)? In our problem, the sample space consists of all ten marbles in the jar, because we are equally likely to draw any one of them.

  14. Probability of drawing a red marble • Using our basic probability fraction, we see that the probability of drawing a red marble at random is:

  15. Classical Probability P(A) = “Probability of Event A.” Classicalprobability is used when each outcome in a sample space is equally likely to occur. The classical probability for event E is given by Example: A die is rolled. Find the probability of Event A: rolling a 5. There is one outcome in Event A: {5}

  16. Toss a coin • What is the probability of getting a head when a coin is tossed? • P( getting a head) =

  17. Gender of Children • If a family has three children, find the probability that all the children are girls.

  18. Expressing Probabilities • When expressing the value of a probability, either: • Give the EXACT fraction or decimal (preferred) • Round off the FINAL decimal result to three significant digits. • In all cases where practical, an exact answer is best. • For example is better than 0.333.

  19. Empirical Probability P(cruise) = Empiricalprobability is based on observations obtained from probability experiments. The empirical frequency of an event E is the relative frequency of event E. Example: A travel agent determines that in every 50 reservations she makes, 12 will be for a cruise. What is the probability that the next reservation she makes will be for a cruise?

  20. Probabilities with Frequency Distributions • Ages • Frequency, f • 13 • 8 • 18 – 25 • 4 • 26 – 33 • 3 • 34 – 41 • 2 • 42 – 49 • 50 – 57 Example: The following frequency distribution represents the ages of 30 students in a statistics class. What is the probability that a student is between 26 and 33 years old? P (age 26 to 33)

  21. Subjective Probability Subjective probability results from intuition, educated guesses, and estimates. This type of probability is not scientific. Example: A business analyst predicts that the probability of a certain union going on strike is 0.15.

  22. Law of Large Numbers Example: Sally flips a coin 20 times and gets 3 heads. The empirical probability is This is not representative of the theoretical probability which is As the number of times Sally tosses the coin increases, the law of large numbers indicates that the empirical probability will get closer and closer to the theoretical probability. As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical (actual) probability of the event.

  23. Complementary Events P (selecting a blue chip) P (not selecting a blue chip) The complement of Event Eis the set of all outcomes in the sample space that are not included in event E. (Denoted .) Example: There are 5 red chips, 4 blue chips, and 6 white chips in a basket. Find the probability of randomly selecting a chip that is not blue.

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