birthday problem what is the smallest number of people you need in a group so that the probability...
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Birthday Problem• What is the smallest number of
people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2?
• Answer: 23No. of people 23 30 40 60Probability .507.706.891.994
Probability
•Formal study of uncertainty•The engine that drives Statistics
• Primary objective of lecture unit 4:
1. use the rules of probability to calculate appropriate measures of uncertainty.
2. Learn the probability basics so that we can do Statistical Inference
Introduction• Nothing in life is certain• We gauge the chances of
successful outcomes in business, medicine, weather, and other everyday situations such as the lottery or the birthday problem
Tomorrow's Weather
A phenomenon is random if individual
outcomes are uncertain, but there is
nonetheless a regular distribution of
outcomes in a large number of repetitions.
Randomness and probabilityRandomness ≠ chaos
Coin toss The result of any single coin toss is
random. But the result over many tosses
is predictable, as long as the trials are
independent (i.e., the outcome of a new
coin flip is not influenced by the result of
the previous flip).
The result of any single coin toss is
random. But the result over many tosses
is predictable, as long as the trials are
independent (i.e., the outcome of a new
coin flip is not influenced by the result of
the previous flip).
First series of tossesSecond series
The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials.
4.1 The Laws of Probability
1. Relative frequencyevent probability = x/n, where x=# of occurrences of event of interest, n=total # of observations– Coin, die tossing; nuclear power plants?
• Limitationsrepeated observations not practical
Approaches to Probability
Approaches to Probability (cont.)
2. Subjective probabilityindividual assigns prob. based on personal experience, anecdotal evidence, etc.
3. Classical approachevery possible outcome has equal probability (more later)
Basic Definitions
• Experiment: act or process that leads to a single outcome that cannot be predicted with certainty
• Examples:1. Toss a coin2. Draw 1 card from a standard deck of
cards3. Arrival time of flight from Atlanta to
RDU
Basic Definitions (cont.)
• Sample space: all possible outcomes of an experiment. Denoted by S
• Event: any subset of the sample space S;typically denoted A, B, C, etc.Null event: the empty set Certain event: S
Examples1. Toss a coin once
S = {H, T}; A = {H}, B = {T}2. Toss a die once; count dots on upper
faceS = {1, 2, 3, 4, 5, 6}A=even # of dots on upper face={2, 4, 6}B=3 or fewer dots on upper face={1, 2, 3}
3.Select 1 card from adeck of 52 cards.S = {all 52 cards}
Coin Toss Example: S = {Head, Tail}Probability of heads = 0.5Probability of tails = 0.5
3) The complement of any event A is the event that A does not occur, written as A.
The complement rule states that the probability
of an event not occurring is 1 minus the
probability that is does occur.
P(not A) = P(A) = 1 − P(A)
Tail = not Tail = Head
P(Tail ) = 1 − P(Tail) = 0.5
Probability rules (cont’d)
Venn diagram:
Sample space made up of an event
A and its complement A , i.e.,
everything that is not A.
Birthday Problem• What is the smallest number of
people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2?
• Answer: 23No. of people 23 30 40 60Probability .507.706.891.994
Example: Birthday Problem
• A={at least 2 people in the group have a common birthday}
• A’ = {no one has common birthday}
502.498.1)'(1)(
498.365
343
365
363
365
364)'(
:23365
363
365
364)'(:3
APAPso
AP
people
APpeople
Mutually Exclusive (Disjoint) Events
• Mutually exclusive ordisjoint events-no outcomesfrom S in common
A and B disjoint: A B=
A and B not disjoint
A
A
Venn Diagrams
Laws of Probability (cont.)
General Addition Rule
5. For any two events A and B
P(A or B) = P(A) + P(B) – P(A and B)
19
For any two events A and B
P(A or B) = P(A) + P(B) - P(A and B)P(A or B) = P(A) + P(B) - P(A and B)
A
B
P(A) =6/13
P(B) =5/13
P(A and B) =3/13
A or B
+_
P(A or B) = 8/13
General Addition Rule
Laws of Probability: Summary
• 1. 0 P(A) 1 for any event A• 2. P() = 0, P(S) = 1• 3. P(A’) = 1 – P(A)• 4. If A and B are disjoint events, then
P(A or B) = P(A) + P(B)• 5. For any two events A and B,
P(A or B) = P(A) + P(B) – P(A and B)
M&M candies
Color Brown Red Yellow Green Orange Blue
Probability 0.3 0.2 0.2 0.1 0.1 ?
If you draw an M&M candy at random from a bag, the candy will have one
of six colors. The probability of drawing each color depends on the proportions
manufactured, as described here:
What is the probability that an M&M chosen at random is blue?
What is the probability that a random M&M is any of red, yellow, or orange?
S = {brown, red, yellow, green, orange, blue}
P(S) = P(brown) + P(red) + P(yellow) + P(green) + P(orange) + P(blue) = 1 P(blue) = 1 – [P(brown) + P(red) + P(yellow) + P(green) + P(orange)]
= 1 – [0.3 + 0.2 + 0.2 + 0.1 + 0.1] = 0.1
P(red or yellow or orange) = P(red) + P(yellow) + P(orange)
= 0.2 + 0.2 + 0.1 = 0.5
Example: toss a fair die once
S = {1, 2, 3, 4, 5, 6}• A = even # appears = {2, 4, 6}• B = 3 or fewer = {1, 2, 3}• P(A orB) = P(A) + P(B) - P(A andB)
=P({2, 4, 6}) + P({1, 2, 3}) - P({2})
= 3/6 + 3/6 - 1/6 = 5/6