Intro to Probability


 Emory McGee
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1 Intro to Probability
2 Random Experiment A experiment is random if: 1) the outcome depends on chance. In other words, the outcome cannot be predicted with certainty (can t know 100%). 2) the set of all possible outcomes is known.
3 Set of possibilities Ex 1: Roll a sixsided die and observe the number on the top side. The set of all possibilities is OR you draw a Venn diagram
4 Equally probable set of possibilities When the possible outcomes of an experiment are equally probable (each outcome has the same probability of occurring), the set of possibilities is called equally probable space. Ex: You roll a die and the chances of either getting a 1, 2, 3, 4, 5, or 6 are all 1 6
5 Vocabulary Simultaneously: happens at the same time. Successively: happens immediately one after another.
6 Single Step vs Several Steps Single Step Several Steps  The roll of a die  The roll of a die followed by the flip of a coin
7 Tree diagram Diagram for random experiment with many steps:
8 Tree diagram example A tree diagram shows all possible meals. At the first step, Kayla chooses a main dish, at the second step, she chooses a dessert. Complete the tree.
9 Tree diagram example The pointers on the two spinners below are turned one after the other.
10 Basic counting rule If 1 event can occur in m ways and a second event can occur in n ways then the number of ways the two events can occur successively is m*n Step 1 # of ways x Step 2 # ways x If we look at the previous example:
11 Basic counting rule example
12 Basic counting rule example The new frozen yogurt shop down the street offers 20 flavors and 8 toppings. You can order a regular, sugar, waffle or chocolate frozen yogurt cone. How many possible ways can you order your frozen yogurt? (assuming that you can only get one type of cone, one flavor of ice cream and one topping for each yogurt cone)
13 Event An event is a part of the set of possibilities. Ex1: When a sixsided die is rolled, rolling an even number is an event that corresponds with the set of outcome {2,4,6} Ex2: When a sixsided die is rolled, rolling a three there is a single outcome in the set of possibilities, that is {3}
14 Types of Events  Impossible An event is...impossible if the probability of occurring is 0 Ex: The event drawing a red marble out of a bag that contains only blue marbles is an impossible event
15 Types of Events  Probable An event is...probable if the probability of occurring is between 0 and 1 Ex: The event drawing a red marble out of a bag that contains blue marbles and red marbles is a probable event
16 Types of Events  Certain An event is...certain if the probability of occurring is 1 Ex: The event drawing a red marble out of a bag that contains only red marbles is a certain event (100% will happen!!!)
17 Incompatible events It is impossible for two events to occur at the same time (simultaneously). Rolling a die Event 1: rolling a # less than 3 Event 2: rolling a # greater than 4.
18 Complimentary events Two events are complimentary if the two events are incompatible and if it is certain that either the 1 st or 2 nd event will occur. Rolling a die Event 1: roll an even # Event 2: roll an odd #
19 Theoretical Probability vs Experimental Probability What is the probability that I will walk outside and get struck by lightning? 1 in 2.or 50% Ummm...that doesn't seem right. Shouldn't it be something like in a million? Well...you either get hit by lightning or you don t. It s that simple I don t get it
20 Theoretical Probability The probability is expressed as a fraction, percent, or even a decimal. What you are looking for
21 Theoretical Probability Ex: When a sixsided die is rolled, the probability of the event rolling an even number is...
22 Theoretical Probability Ex: When a sixsided die is rolled, the probability of the event rolling a 3 is...
23 Experimental Probability The probability of an event obtained as a result of experiment (or observation). This is often used when theoretical probability cannot be calculated or when it makes more sense (like the probability of being struck by lightning.)
24 Experimental Probability The greater the number of repetitions of an experiment, the closer the experimental probability will be to the theoretical probability.
25 Or Means to Add When you are determining to the probability of A or B, you simply add the probability of A and the probability of B. This is true as long as A and B cannot happen at the same time.
26 Probability of an Event Ex: If a bag contains 4 blue marbles, 3 red marbles and 6 green marbles, the probability of the event drawing a blue or red marble is given by:
27 Sum of Probabilities = 1 Ex: If a bag contains 4 blue marbles, 3 red marbles and 6 green marbles, the probability of the event drawing a blue, red or green marble is 1. You simply add up all the probabilities See next slide.
28 Sum of Probabilities = 1 A certain event
29 Experiments Involving Several Steps If a random experiment involves several steps, the probability of an event is equal to the product of the probabilities of each of the events in each step. Event 1 occurs m ways Event 2 occurs n ways Total outcome = m*n
30 And or Followed by Means to Multiply When you are determining to the probability of A and B, or A followed by B, multiply the probability of A and the probability of B
31 Experiments Involving Several Steps  WITHOUT Replacement Ex: If a bag contains 4 blue marbles, 3 red marbles and 6 green marbles, the probability of the event drawing a blue marble followed by a red marble without replacing the blue (don t put a blue marble back) is See next slide
32 Experiments Involving Several Steps  WITHOUT Replacement
33 Experiments Involving Several Steps  WITH Replacement Ex: If a bag contains 4 blue marbles, 3 red marbles and 6 green marbles, the probability of the event drawing a blue marble followed by a red marble after replacing the blue is See next slide
34 Experiments Involving Several Steps  WITH Replacement
35 Dependent and independent events Two events A and B are independent in probability when the occurrence of one event does not influence the probabiity of the other occuring. Otherwise the events are called dependent. Ex: Event A choosing a white ball on the first drawing and event B choosing a white ball on the second drawing are independent when drawing with replacement and dependent when drawing without replacement.
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10.3 TEKS a.1, a.4 Define and Use Probability Before You determined the number of ways an event could occur. Now You will find the likelihood that an event will occur. Why? So you can find reallife geometric
More informationPERCENT PROBLEMS USING DIAGRAMS and 5.1.2
PERCENT PROBLEMS USING DIAGRAMS 5.1.1 and 5.1.2 A variety of percent problems described in words involve the relationship between the percent, the part and the whole. When this is represented using a number
More informationA 21.0% B 34.3% C 49.0% D 70.0%
. For a certain kind of plant, 70% of the seeds that are planted grow into a flower. If Jenna planted 3 seeds, what is the probability that all of them grow into flowers? A 2.0% B 34.3% C 49.0% D 70.0%
More informationIf a regular sixsided die is rolled, the possible outcomes can be listed as {1, 2, 3, 4, 5, 6} there are 6 outcomes.
Section 11.1: The Counting Principle 1. Combinatorics is the study of counting the different outcomes of some task. For example If a coin is flipped, the side facing upward will be a head or a tail the
More informationSERIES Chance and Probability
F Teacher Student Book Name Series F Contents Topic Section Chance Answers and (pp. Probability 0) (pp. 0) ordering chance and events probability_ / / relating fractions to likelihood / / chance experiments
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