Probability Concepts

• Probability & Statistics for Engineers & Scientists, Ninth Edition, is written by Ronald E. Walpole, Raymond H. Myers, and others.

Random (Statistical) Experiment

• An experiment with known outcomes but unpredictable results before execution.
• Examples of random experiments:
  • The roll of a dice
  • The toss of (flipping) a coin

Sample Space (S)

• Represents the set of all possible outcomes of a statistical experiment.
• For the roll of a dice: S = {1,2,3,4,5,6}
• For flipping a coin: S = {Head,Tail} or S = {H,T}

Tree Diagrams

• Sample spaces are described graphically with tree diagrams.

Sample Space Examples

• Flipping a coin twice results in the sample space S = {HH, HT, TH, TT}.
• Selecting three items from a manufacturing process, where each item is classified as either defective (D) or nondefective (N), the sample space is S = {DDD, DDN, DND, DNN, NDD, NDN, NND, NNN}, with n(S) = 23 = 8.
• An experiment includes flipping a coin; if heads, it is flipped a second time, or if tails, a die is tossed: S = {HH, HT, T1, T2, T3, T4, T5, T6}
• A dice is rolled twice and the event that the sum of the faces is greater than 7, given that the first outcome was a 4 is: S = {11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66} E = {44, 45, 46}
• In an experiment involving tossing a pair of dice, the sample space S consists of all possible pairs: S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}, where n(S) = 62 = 36.
• By tossing a coin twice, S = {HH, HT, TH, TT}. n(S) = 4, where X is a random variable which represents the number of heads: X = {0, 1, 2}

Discrete Sample Space

• If a sample space contains a finite number of possibilities(or unending sequence), it is a discrete sample space.

Continuous Sample Space

• If a sample space contains an infinite number of possibilities, it is a continuous sample space.
• Interest centers around the proportion of people who respond to a mail solicitation, X is on all values x for which 0 ≤ x ≤ 1.
• Example of a continuous sample space is cell phone camera recycle times.
• Given S = R+ = {x | x > 0}
• All recycle times are between 1.5 and 5 seconds the sample space can be S = {x | 1.5 < x < 5}

Random Variable

• A function that associates a real number with each element in the sample space.
• A random variable on a continuous scale is called a continuous random variable.
• Discrete Variable: X = {0, 1, 2}
• Continuous Variable: 0 ≤ X ≤ 1

Discrete and Continuous Random Variables

• Two items are selected from a manufacturing process and classified as defective (D) or nondefective (N).
• Where X represents the number of N,
• If X=0, Outcome is DD
• If X=1, Outcomes are DN and ND
• If X=2, Outcome is NN

Example for classifying defective and non-defective

• Three items are selected from a manufacturing process and classified as D or N.
• Where X represents the number of N:
  • S = {DDD, DDN, DND, DNN, NDD, NDN, NND, NNN}.
• In tossing a coin twice:
  • S = {HH, HT, TH, TT}.
  • n(S) = 4.

Discrete Probability Distributions Function (pdf)

• The set of ordered pairs (x, f(x)) is a probability function, probability mass function, or probability distribution of the discrete random variable X if, for each possible outcome x:
  • f(x) ≥ 0
  • Σ f(x) = 1
  • P(X = x) = f(x)

Flipping a coin twice for probability distributions

• S = {HH, HT, TH, TT}. n(S) = 4.
• X symbolizes a random variable which represents the number of appeared heads: X= {0, 1, 2}
• f(0) = ¼
• f(1) = ½
• f(2) = ¼
• Graph the discrete probability distribution for f(x) where axes are x and f(x) and coordinates are (0, ¼), (1, ½), and (2, ¼)
• Flipping a coin 3 times:
  • S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.
  • n(S) = 8. X is random variable symbol and the of heads
  • X = {0, 1, 2, 3} where x and f(x) have results:
    • f(0) = ⅛
    • f(1) = ⅜
    • f(2) = ⅜
    • f(3) = ⅛
• Graph discrete probability distribution where coordinates for the random variable
  • X= {0, 1, 2, 3} are the horizontal axis and the distribution is f(x)

Discrete Cumulative Distribution Function (cdf)

• Is defined with the formula F(x) = P(X ≤ x) = Σ f(t) for -∞ < x < ∞.
• For tossing a coin twice: S = {HH, HT, TH, TT}, n(S) = 4 and X is a random variable which represents the # of heads. X= {0, 1, 2}
  • f(0) = ¼
  • f(1) = ½
  • f(2) = ¼ and then the distribution function for F(x)

Discrete Cumulative Distribution Function example

• For tossing a coin 3 times
  • S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.
  • n(S) = 8,
  • X= {0, 1, 2, 3} where f(x) displays distribution for totals: where f(0) displays total distribution of ⅛, f(1)displays ⅜, f(2) equals ⅜ and f(3) equaling ⅛
    • F(x) is illustrated by 0, x0
    • 1, f(1) +f(2) + f(3) + f(0), x≥3
• Direct calculations of the probability distribution
  • f(0)= 1/16, f(1) = 1/4, f(2)=3/8, f(3)= 1/4, and f(4)= 1/16
• If f(x) = (1/16) * (4/x), where x has a set x = 0,1,2,3,4 the solutions are:
  • F(0)=1/16
  • F(1)=5/16
  • F(2) = 1/16
• f(2) = F(2) - F(1) = 3/8
• When determining the value of c for a probability distribution function of a discrete random variable X: If for x = 0, 1, 2, 3; f(x) = c(x² + 4):
• Where f(0) = 4c, f(1) = 5c, f(2) = 8c, and f(3) = 13c Sum = 30c
• 30c = 1

The cumulative distribution function of X

• Is the number of imperfections per 10 meters of synthetic fabric.
• By creating functions for:
• Where probability for function F(x) = 0 for x0,1,2,3,4 and the distribution for continuous rolls of equal width shows graph of sample.
• A discrete random variable X has the probability distribution, calculate F(3), when P(X = x) = x/15 is in sets x is an element {of 1, 2, 3, 4, 5}.
  • By taking data when P ( X ≤ 3) = P(X = 1) + P(X = 2) + P(X=3), we find P ( X ≤ 3) = ⅔.