statistical reasoning for everyday life
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Statistical Reasoning for everyday life. Intro to Probability and Statistics Mr. Spering – Room 113. 6.5 Combining Probabilities. Find the probability. What is the probability of rolling 2 or a 5 on a number cube? 2/6 or 33.33% - PowerPoint PPT PresentationTRANSCRIPT
Statistical Reasoningfor everyday life
Intro to Probability and Statistics
Mr. Spering – Room 113
6.5 Combining Probabilities Find the probability.
What is the probability of rolling 2 or a 5 on a number cube? 2/6 or 33.33% A bag contains 32 red marbles, 30 blue marbles, and 18 white
marbles. You pick one marble from the bag. Find P (picking blue).
3/8 or 37.5% P (not a red) 3/5 or 60% What is the probability of having a sample with mean age
between 35 years and 45 years, given the population mean is 40 years and the standard of deviation is 2.5 years?
95% Using a regulation deck of cards. What is the probability of
choosing a Queen of Hearts? 1/52, 0.019, or 1.9%
6.5 Combining Probabilities Contingency Tables
Tree Diagrams
Red 2 24 26
Black 2 24 26
Total 4 48 52
Ace Not Ace Total
Full Deck of 52 Cards
Red Card
Black Card
Not an Ace
Ace
Ace
Not an Ace
Sample Space
Sample Space2
24
2
24
6.5 Combining Probabilities
Venn DiagramsLet A = acesLet B = red cards
A
B
A ∩ B = ace and red
A U B = ace or red
6.5 Combining Probabilities PERMUTATIONS = Arrangements (Order matters) Permutations: The number of ways of arranging X objects
selected from n objects in order is
Example: Your restaurant has five menu choices, and three are selected
for daily specials. How many different ways can the specials menu be ordered?
Answer: different possibilities
X)!(n
n!Pxn
602
120
3)!(5
5!
X)!(n
n!nPx
6.5 Combining Probabilities COMBINATIONS = Grouping (Order does not matter) Combinations: The number of ways of selecting X
objects from n objects, irrespective of order, is
Example: Your restaurant has five menu choices, and three are
selected for daily specials. How many different special combinations are there, ignoring the order in which they are selected?
Answer: different possibilities
X)!(nX!
n!Cxn
10(6)(2)
120
3)!(53!
5!
X)!(nX!
n!Cxn
6.5 Combining Probabilities Joint Probabilities (AND probabilities) Independent VS. Dependent… Independent events are events where the outcomes of one does
not affect the outcomes of another. Dependent events are events where the outcome of one will affect the outcome of another.
Independent → flipping a coin
Dependent → Drawing two cards after drawing a card
6.5 Combining Probabilities Independent…
AND probability… Considering two independent events A and B
that have individual probabilities P(A) and P(B). The probability that A and B occur together is:
Concept may be extended for more than 2 events.
)()() and ( BPAPBAP
)()()() and and ( CPBPAPCBAP
6.5 Combining Probabilities Independent…Example
Suppose you have a coin and a spinner with 5 equal sectors, labeled 1 thru 5. What is the probability of spinning an even number AND getting heads?
)()() and ( BPAPBAP 1 2 2 1
( and )2 5 10 5
P heads even
6.5 Combining Probabilities Dependent …{Conditional Probability}
AND probability… Considering two events A and B. The
probability that A and B occur together is:
Concept may be extended for more than 2 events.
)given ()() and ( ABPAPBAP
) and given ()given ()() and and ( BACPABPAPCBAP
6.5 Combining Probabilities Dependent …
The game of BINGO involves drawing pieces with a letter and a number on each piece. If we draw at random without replacement. Find the probability of drawing two B pieces in the first two selections, given there are 75 pieces, 15 for each of the letters B, I, N, G, O!
)given ()() and ( ABPAPBAP
0378.0185
7
74
14
75
15B) and B( P
6.5 Combining Probabilities
)()()or ( BPAPBAP
•Either/OR Probability: [Disjunction]
NON-OVERLAPPING EVENTS…
Two events that can not occur at the same time, the probability that either A or B occurs is
Concept may be extended for more than 2 events.
)()()()or or ( CPBPAPCBAP
6.5 Combining Probabilities•Either/OR Probability… Example…
NON-OVERLAPPING EVENTS…
What is the probability of rolling a die and getting a 3, 4, or 7?
3
1
6
2
6
0
6
1
6
1)7or 4or 3( P
6.5 Combining Probabilities•Either/OR Probability:
OVERLAPPING EVENTS…
When two events are considered either/or, but may occur at the same time, then the probability that A or B occurs is:
) and ()()()or ( BAPBPAPBAP
6.5 Combining Probabilities
) and ()()()or ( BAPBPAPBAP
MEN WOMEN
American 2 6
French 4 8
•Either/OR Probability:
OVERLAPPING EVENTS…Consider this situation on tourism…Given the table, what is the probability of meeting at random a person who is either a woman or French?
%9020
18
20
8
20
12
20
14)or ( FrenchwomanP
6.5 Combining Probabilities
AND probability
Independent events
AND probability Dependent
events
Either/OR probability
Non-overlapping
events
Either/OR probability
Overlapping events
)given ()(
) and (
ABPAP
BAP
)()(
)or (
BPAP
BAP
•Summary of Combining Probabilities:
•QUESTIONS????
) and ()()(
)or (
BAPBPAP
BAP
)()(
) and (
BPAP
BAP
6.5 Combining Probabilities
HOMEWORK: pg 274 # 1-27 all