Intro to Probability

Transcription

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 six-sided 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 six-sided die is rolled, rolling an even number is an event that corresponds with the set of outcome {2,4,6} Ex2: When a six-sided 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 six-sided die is rolled, the probability of the event rolling an even number is...

22 Theoretical Probability Ex: When a six-sided 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.