Chapter 4 Probability Distributions Sections 4.14.4

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Chapter 4Probability Distributions(Sections
4.1-4.4)

• MATH320

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Section 4.1Overview
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Overview

• Deals with the construction of probability
  distributions
• Combine methods from chapters 2 and 3
• Descriptive Statistics (the actual experiment)
• Probability (find possible outcomes)
• Probability distributions will
• Discuss what will happen
• Not what has happened

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What are these numbers?
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How about now?
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Lets Distinguish Between Them?
Last Digit Represents Set A Measured
Weights Set B Asking Weights
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Data Statistics
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Probability Distributions
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Section 4.2Random Variables
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Definitions

• A random variable is a variable (typically
  represented by x) that has a single numerical
  value, determined by chance, for each outcome of
  a procedure.
• A probability distribution is a graph, table, or
  formula that gives the probability for each value
  of the random variable.

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Definitions

• Discrete Random Variable has
• Finite number of values
• Countable number of values
• Example a counter
• Continuous Random Variable has
• Infinitely many values
• associated with measurements on a continuous
  scale
• With no gaps or interruptions
• Example a scale
• To determine, choose potential values. If value
  could be between other values, its continuous

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Examples of Discrete Random Variables
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Examples of Continuous Random Variables
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Graphs

• Similar to distribution frequencies
• Uses probability along y-axis
• Not the count or frequency
• Has to add up to 100 or else not normally
  distributed
• ?P(x)1
• 0P(x)1

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Probability Distributions

• Describes how probabilities are distributed over
  the values of the random variable
• Variable x
• Probability P(x)
• Probability of each variable is included
• P(1)0.39
• P(3 or more)0.140.040.010.19

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Example (4 6 p. 192)
?P(x)0.977 ? 1.00 Not normally distributed
?P(x)1.0000 Normally distributed
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Important Characteristics of Data

• Center
• Variation
• Distribution
• Outliners
• Time

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Formulas
Must be normally distributed to use.
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Range Rule Applies

• Most values should fall between of the standard
  deviation of the sample.
• We can therefore identify unusual values by
  determining if they lie outside these
  limits Maximum usual value µ 2s Minimum
  usual value µ 2s
• µ mean (population)
• s standard deviation (population)

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Observed Dice Roll
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Example (7 p. 193)
? 0.73 ?2 1.324 (0.73)2 0.791 ? 0.889
(square root of 0.791)
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Round Off Rule
Round results by carrying one more decimal place
than the number of decimal places used for the
random variable x. If the values of x are
integers, round µ, ?, ?and ?2 and to one decimal
place. There are exceptions when the precision
of the number is important.
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Unusual Results

• Under a given assumption (a coin is fair)
• Probability of a particular event (992 heads in
  100 tosses) is extremely small
• Conclude the assumption is probably not correct

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Using Probabilities unusual results

• Successes (x) among trials (n)
• Unusually high P(x or more)0.05
• Unusually low P(x or more) 0.05
• Both are unusual

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Section 4.3Binomial Probability Distributions
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Binomial Experiment

• Information obtained from a probability
  experiment
• Experiment must satisfy the following
• Each trial can only have TWO outcomes
• Success outcome (p)
• Failure outcome (q)
• Fixed number of trials
• Outcomes are independent of each other
• Probability for success must remain the same

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Binomial Examples

• Flip three coins
• Break rule? NO
• Each coin has only two possible outcomes
• Roll a die
• Doesnt fit as a whole
• Too many outcomes
• Can be considered when looking at the outcome of
  a FIVE
• Hitting a target

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Notation for Binomial Prob Distribution

• Success (S) and Failure (F)
• P(S)p
• P(F)1-pq
• n denotes the fixed number of trials
• x number of successes
• p probability of success
• q probability of failure
• P(x) probability of getting exactly x success
  among the n trial

x and p must represent the same category
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Important Hints

• Be sure that x and p both refer to the same
  category being called a success.
• When sampling without replacement, the events can
  be treated as if they were independent if the
  sample size is no more than 5 of the population
  size. (That is n 0.05N.)

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Methods For FindingProbabilities
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Method 1 Using Binomial Formula
Where n number of trials x number of
successes among n trials p probability of
success in any one trial q probability of
failure in any one trial (q 1 p)
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Rational
n !
P(x) px qn-x
(n x )!x!
Number of outcomes with exactly x successes
among n trials
Probability of x successes among n trials for any
one particular order
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Example

• An archer hits the bulls eye 80 of the time. If
  he shoots 5 arrows, find the probability that he
  will get 4 bulls eyes.
• n5
• x4
• p0.80
• q0.20
• Break math into three groups

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Method 2 Table A-1 in Appendix A
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Method 3 Use Technology
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Using Minitab
Set up C1 with the x values (0, 1, 2, 3, 4)
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Using Excel
Must use false as the cumulative value
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Strategy

• Use computer software or a TI-83 calculator if
  available.
• If neither software or the TI-83 Plus calculator
  is available, use Table A-1, if possible.
• If neither software nor the TI-83 Plus calculator
  is available and the probabilities cant be found
  using Table A-1, use the binomial probability
  formula.

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Examples

• Student takes a 5-question true-false quiz. Since
  the student hasnt studied, he decides to flip a
  coin to determine the correct answers. What is
  the probability that the student guesses exactly
  3 out of 5 correctly?
• n5, x3, p0.50 P(exactly 3 correct)0.3125
• A circuit has 6 breakers. The probability that
  each breaker will fail is 0.1. If the circuit is
  activated, find the probability that exactly two
  breakers will fail. Each breaker is independent
  of the other
• n6, x2, p0.1 (P2 will fail)0.098415

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Example -- 28 p. 205

• Getting one wrong number P(x1)0.347

• Getting at least one wrongP(x1)P(x0)P(X1)
  0.3470.1970.544

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Affirmative Action Program Example

• A study was conducted to determine whether there
  were significant differences between medical
  students admitted through special programs
  (affirmative action, etc) and medical students
  admitted through the regular admissions criteria.
  It was found that the graduation rate was 94 for
  the medical students admitted through special
  programs (source Journal of the American Medical
  Association).

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Affirmative Action Program Example (cont)

• If 10 students from the special program are
  randomly selected, find the probability that at
  least 9 of them graduated
• P(x?9) P(x9) P(x10)
• n10
• p0.94
• x9 and x10
• P(x?9)0.8824

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Authors Slot Machine

• The author purchased a slot machine that is
  configured so that there is a 1/2000 probability
  of winning the jackpot on any individual trial. A
  guest claims that she played the slot machine 5
  times and hit the jackpot twice.
• Probability of exactly two jackpots in five
  trials
• P(x2)
• Probability of at least two jackpots in five
  trials
• P(x?2)P(2)P(3)P(4)P(5)
• Guests claim seem valid?
• No, extremely unlikely

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Using Data to Answer Questions

• Probability that at least five of the subjects
  experienced headaches P(x?5) P(5)P(6)
• Probability that at most two subjects experienced
  headaches P(x2) P(0)P(1)P(2)
• Probability that more than one subject
  experienced headaches P(xgt1)
  P(2)P(3)P(4)P(5)P(6)
• Probability that at least one subject experienced
  headaches P(1)P(2)P(3)P(4)P(5)P(6) P(x?1)
  1-P(0)

n6 p0.167
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Section 4.4Mean, Variance, and Standard
Deviation for Binomial Dist
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Any Discrete Probability Distribution Formulas
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Binomial Distribution Formulas
Where n number of fixed trials p probability
of success in one of the n trials q probability
of failure in one of the n trials
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Interpreting the Results
It is especially important to interpret results.
The range rule of thumb suggests that values are
unusual if they lie outside of these limits
Maximum usual values µ 2 ? Minimum usual
values µ 2 ?
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Examples

• Twelve cards are selected from a deck and each
  card is replaced before the next one is drawn.
  Find the average number of diamonds.
• n12
• p13/53 or 1/4 or 0.25
• ? n p
• 12 0.25 3
• Interpretation on average, we would expect to
  draw three diamonds in twelve draws.

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Example

• A die is rolled 180 times. Find the standard
  deviation of the number of threes.
• n180
• p1/6
• q5/6