PROBABILITY Review of Unit 1; 8th grade
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Probability
Enduring Understandings : Tree diagrams are useful for describing small spaces and finding outcomes in a visual manner . When one outcome does not effect another they are independent events. When event A happens AND event B happens we multiply to find the probability When we want to find the probability of event A OR event B we add. Probabilities can be written as percents, decimals or ratios. The sum of the probabilities of every outcome = 1.
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Discussion I) Theoretical vs. Experimental A) conduct experiments B) On paper II) Fair vs Unfair games A) Equal outcomes for all players B) unequal outcomes for all players III) And/ Or A) and= multiply; and backwards =do not add B) or = add IV. Mutually Exclusive vs. Mutually Inclusive A) Events cannot happen together B) Events can happen together
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Problems Involving Probability A) Single & Compound Events B) Probability Involving AND & OR C) Find combinations and outcomes 4
Single Events A single event involves the use of ONE item such as: one card being drawn one coin being tossed one die being rolled one person being chosen 5
Application Time Example: From a normal deck of 52 cards, what is the probability of choosing the queen of clubs? The deck contains only one queen of clubs, so the probability will be 1/52. 6
Compound Events A compound event involves the use of two or more items such as: Two cards being drawn Three coins being tossed Two dice being rolled Four people being choosen 7
Application Time Example: How many different 4 letter words can be formed from the letters in the word MATH? Choices go down after being used once: 4 • 3 • 2 • 1 = 24 8
Work Time Exercises:
1) Which of the following illustrates working with compound events? A) rolling a die B) tossing 2 coins C) drawing one card D) choosing one person
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A standard deck of 52 cards is shuffled. What is the probability of choosing the 5 of diamonds?
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Probability involving AND & OR In probability, an outcome is in event ‘A and B’ only when the outcome is in both event A and event B.
P( A and B)= n( A and B)/ n(S) *n(A and B):# of outcomes in both A and B. *n(S): the total # of possible outcomes 10
Application Time Example: A die is rolled. What is the probability that the number is even and less than 4? Event A: Numbers on a die that are even: 2, 4, 6 Event B: Numbers on a die that are less than 4: 1, 2, 3 There is only one number (2) that is in both events A and B. Total outcomes S: Numbers on a die:1, 2, 3, 4, 5, 6 Answer:Probability = 1/6 11
Probability involving AND & OR In probability, an outcome is in event ‘A or B’ when the outcome is in either (or both) event A or event B.
P(A or B)=P(A)+P(B)-P(A and B) 12
Application Time Example: A die is rolled. What is the probability that the number is even or less than 4? Event A: Numbers on a die that are even: 2,4,6 P(A)=3/6 Event B: Numbers on a die that are less than 4: 1, 2, 3 P(B)=3/6 P(A and B) = 1/6 Answer: Probability = P(A) + P(B) - P(A and B) = 3/6 + 3/6 - 1/6 = 5/6 13
Work Time Exercises: 1)A die is rolled. What is the probability that the number rolled is greater than 2 and even?
2) A standard deck of cards is shuffled and one card is drawn. Find the probability that the card is a queen or an ace. 14
II) COMPUTING PROBABILITIES
A) Mutually Exclusive and Independent Events B) Counting Principle C) Sample Space
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A) Mutually Exclusive and Independent Events
Complement of an Event Mutually Exclusive Events Independent Events
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Complement of an Event If A is an event within the sample space S of an experiment, the complement of A (denoted A’) consists of all outcomes in S that are not in A. The complement of A is everything else in the problem that is NOT in A.
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Application Time Experiment: Tossing a coin Event
A
The coin shows heads
Complement
A’
The coin shows tails
Experiment: Drawing a card Event
A
The card is black
Complement
A’
The card is red
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Complement of an Event The probability of complement of an event is one minus the probability of the event.
P(A’)=1-P(A)
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Application Time Example: A pair of dice are rolled. What is the probability of not rolling doubles? P(doubles) = 6/36 = 1/6 P(not doubles) = 1 - 1/6 = 5/6
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Mutually Exclusive Events Two events that have NO outcomes in common are called mutually exclusive.These are events that cannot occur at the same time. Think of this as the 2 events together (mutually) agreeing to exclude (not include) each others' elements.  They have agreed to be different - mutually exclusive. 21
Application Time Example: A pair of dice is rolled. The events of rolling a 9 and of rolling a double have NO outcomes in common. These two events ARE mutually exclusive.
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Mutually Exclusive Events For any two mutually exclusive events, the probability that an outcome will be in one event or the other event is the sum of their individual probabilities. If A and B are mutually exclusive events, P(A or B) = P(A) + P(B) 23
Mutually Exclusive Events For any two events which are not mutually exclusive, the probability that an outcome will be in one event or the other event is the sum of their individual probabilities minus the probability of the outcome being in both events. Look out!! Don't get stuck on this one!!! If events A and B are NOT mutually exclusive, P(A or B) = P(A) + P(B) - P(A and B)
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Application Time Example: A pair of dice is rolled. What is the probability that the sum of the numbers rolled is either 7 or 11? Six outcomes have a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) P(7) = 6/36 Two outcomes have a sum of 11: (5,6), (6,5) P(11) = 2/36 The sum of the numbers cannot be 7 and 11 at the same time, so these events are mutually exclusive. P(7 or 11) = P(7) + P(11) = 6/36 + 2/36 = 25 8/36 = 2/9
Independent Events
Two events are said to be independent if the result of the second event is not affected by the result of the first event.
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Independent Events ď‚Ź If A and B are independent events, the
probability of both events occurring is the product of the probabilities of the individual events If A and B are independent events P(A and B) = P(A) x P(B). 27
Application Time Example: A drawer contains 3 red paperclips, 4 green paperclips, and 5 blue paperclips. One paperclip is taken from the drawer and then replaced. Another paperclip is taken from the drawer. What is the probability that the first paperclip is red and the second paperclip is blue? Because the first paper clip is replaced, the sample space of 12 paperclips does not change from the first event to the second event. The events are independent. P(red then blue) = P(red) x P(blue) = 3/12 • 5/12 = 15/144 = 5/48. 28
Independent Events
If the result of one event IS affected by the result of another event, the events are said to be dependent.
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Independent Events If A and B are dependent events, the probability of both events occurring is the product of the probability of the first event and the probability of the second event once the first event has occurred. If A and B are dependent events, and A occurs first, Â P(A and B) = P(A) x P(B,once A has 30 occurred)
Application Time Example: A drawer contains 3 red paperclips, 4 green paperclips, and 5 blue paperclips. One paperclip is taken from the drawer and is NOT replaced. Another paperclip is taken from the drawer. What is the probability that the first paperclip is red and the second paperclip is blue?
Because the first paper clip is NOT replaced, the sample space of the second event is changed. The sample space of the first event is 12 paperclips, but the sample space of the second event is now 11 paperclips. The events are dependent. P(red then blue) = P(red) x P(blue) = 3/12 • 5/11 = 15/132 = 5/44.
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Work Time Answer the following questions dealing with mutually exclusive,independent events,and complements of events. 1) A pair of dice is rolled.Two possible events are rolling a number greater than 8 and rolling an even number. Are these two events mutually exclusive events?
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Work Time(Following) 2) A pair of dice is rolled. A possible event is rolling a multiple of 5. What is the probability of the complement of this event? 3) A paper bag contains 15 slips of paper. Eight of them are blue and are numbered from 1 to 8. Seven of them are pink and are numbered from 1 to 7. What is the probability of drawing a slip of paper with an even number? 33
Counting Principle The Counting Principle works for two or more activities. For example, if ice cream sundaes come in 5 flavors with 4 possible toppings, how many different sundaes can be made with one flavor of ice cream and one topping? Rather than list the entire sample space with all possible combinations of ice cream and toppings, we may simply multiply 5 • 4 = 20 possible sundaes. This simple multiplication process is known as the Counting Principle. 34
Application Time Example:A coin is tossed five times. How many arrangements of heads and tails are possible? By the Counting Principle, the sample space (all possible arrangements) will be 2•2•2•2•2 = 32 arrangements of heads and tails.
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Sample Space A sample space is a set of all possible outcomes for an activity or experiment. Activity Rolling
Sample Space {1, 2, 3, 4, 5, 6}
a die Tossing
{ Heads, Tails}
a coin 36
Sample Space
Remember:Â A sample space is the set of ALL possible outcomes.
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Work Time Exercises:
1) Alarm clocks are sold in blue or pink with either digital or standard displays. How many different arrangements of alarm clocks are possible? List the sample space.
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Work Time 2) A movie theater sells 3 sizes of popcorn (small, medium, and large) with 3 choices of toppings (no butter, butter, extra butter). How many possible ways can a bag of popcorn be purchased? 39
Work Time A password requires 3 letters and 2 numbers followed by a constant. If the letters are not case sensitive and the digits 0-9 can be used for the numbers, how many combinations of passwords can be made.
26x26x26x10x10x21=
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Test Time ď‚ŹStudy all of these concepts and
practice the problems and you will do very well on the test tomorrow.
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