4. Probability

 

Relative Frequency Definition of Probability

The relative frequency definition of probability says: If an experiment is repeated, the probability of an event (a specified outcome of the experiment) is the relative frequency of occurrence of that event in a large number of repetitions of the experiment.

In this definition 'large number' is vague and will not be made precise here.

The following links use the Virtual Laboratories in Probability and Statistics by Professor Kyle Siegrist of the University of Alabama at Huntsville.  Each link opens a page that illustrates the relative frequency definition of probability for a different situation.

  • The Birthday Problem

A page by Professor Siegrist with a link to a simulation of the birthday problem is found here.  When you reach this page, scroll down until you reach red underlined text that says birthday experiment.  Follow this link to open the birthday problem simulation.  Set the number of people in a room at 30 (k=30).  The height of the blue bar on the right above 0.0 indicates the probability of no matching birthdays while the bar above 1.0 shows the probability of at least two of the people having the same birthday.  What are these probabilities for 30 people?  What are they if the number of people in the room is set at 20?   At 23?  Now set k at 30 and run the simulation of the experiment 100 times.  What is the actual proportion of cases in which there were no matching birthdays?  What is the actual proportion of cases in which there was at least one matching birthday?

 

  • Tossing Coins

A link to a simulation of coin tossing is found here.  When you reach this page, scroll until you see coin experiment underlined in red.  Follow this link to open the coin tossing simulation.  Run the coin tossing simulation with 1 single fair coin 100 times.  Record the number of times a head occurs in the 100 tosses.   How many heads would you expect?  Does the number that you have observed lead you to believe that the coin is not fair?

Now run the experiment with 2 fair coins.  How many times would you expect to get 2 heads?  What did you observe in your 100 tosses?

 

  • Tossing Dice

Follow the link in the previous paragraph and scroll to the red-underlined dice experiment on the linked page.  Run the die tossing simulation with 1 fair die 100 times.  Record the number of times that a 6 occurs.  For a fair die, about how many times would you expect a 6 to occur?  Now change the experiment so 2 fair dice are tossed.  About how many times would you expect a sum of 5 on the dice to occur in 100 tosses?  How many times did you get a sum of 5 on your 100 tosses?

 

  • Drawing Cards

From the same linked page in the last two sections scroll to the red-underlined card experiment and follow this link.  Run the card experiment 20 times.  About how many times should a red card occur in 20 runs of the experiment?  How many times did you get a red card in your 20 runs of the experiment?

This link goes to a page on drawing cards for a Poker Deck.  When the page opens follow the red Poker Experiment link  Choose a certain outcome.  Try a few hands and see if the actual probabilities of your chosen outcome are close to the theoretical outcomes of that event.

 

  • The Matching Experiment

The matching experiment is explained at this link.  Go to this link, read information on the matching experiment, and try the experiment by pressing matching experiment underlined in red.

 

The above examples involve gambling.  The mathematical study of probability has roots in the study of certain gambling situations.  The next link takes you to a web page that has much information about modern gambling.  If you choose to visit the page be careful if you don't want to get on their mailing list.  The page is called the Wizard of Odds .

 

Sample Space Definition of Probability

  • Experiment

    An experiment can be repeated and has clearly defined and identifiable results.  In the previous section five different experiments are outlined.

  • Sample Point

    Any fundamental outcome of an experiment.


  • Sample Space

    The collection of all sample points for an experiment.


  • Event

    Any subset of the sample space of an experiment.  Events are usually denoted by capital letters from the first part of the alphabet.


  • Probability of an Event

    • The probability of an event is a number between 0 and 1 (it may be 0 or 1).  The probability of event A is denoted by P[A].
    • Probability indicates the likelihood that the event will occur.  Events with probabilities close to 1 are more likely to happen than events with probabilities near 0.


  • Probability Laws

    • P[Not E]=1-P[E]
    • P[A or B]=P[A]+P[B]-P[A and B]

  • Examples

    • Example 1: A fair coin is tossed 3 times.  Events defined are: A=At least one head in the 3 tosses, B=Exactly 2 heads in the 3 tosses, and C=No heads in 3 tosses.  Find P[A], P[B], P[C], P[Not A], P[Not B], P[Not C], P[A or B], P[A or C], P[B or C], P[A and B], P[A and C], P[B and C].

      • The sample space consists of sample points HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT.  These sample points can be divided into those in A (red ones), those in B (underlined ones), and those in
        C (italicized ones).  They are then:

      HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT

      • From the last display, A consists of sample points in red, so P[A]=7/8; the sample points in Not A are those not red, so P[Not A]=1/8.

      • From the last display, B consists of sample points underlined, so P[B]=3/8; sample points in Not B are those not underlined, so P[Not B]=5/8.

      • From the last display, C consists of sample points italicized, so P[C]=1/8; sample points in Not C are those not italicized, so P[Not C]=7/8.

      • From the last display, A and B is all sample points that are both red and underlined,
        so P[A and B]=3/8 while A or B is all sample points that are red or underlined (or both),
        so P[A or B]=7/8.  Notice that P[A or B]=7/8=P[A]+P[B]-P[A and B]=7/8 + 3/8 - 3/8.

      • From the last display, A and C is all sample points that are both red and italicized,
        so P[A and C]=0/8 while A or C is all sample points that are red or italicized (or both),
        so P[A or C]=8/8.  Notice that P[A or C]=8/8=P[A]+P[C]-P[A and C]=7/8 + 1/8 - 0/8.

      • From the last display, B and C is all sample points that are both underlined and italicized,
        so P[B and C]=0/8 while B or C is all sample points that are underlined or italicized (or both),
        so P[B or C]=4/8.  Notice that P[B or C]=4/8=P[B]+P[C]-P[B and C]=3/8 + 1/8 - 0/8.


    • Example 2: A pair of fair dice are tossed.  One die is green and the other die is white.  Events defined are: A=Green die is greater than or equal to 2, B=White die shows an even number, C=Sum of numbers on the dice is 6. Find P[A], P[B], P[C], P[Not A], P[Not B], P[Not C], P[A or B], P[A or C], P[B or C], P[A and B], P[A and C], P[B and C].

      The number on the white die is shown on the left margin of the next table, and the number on the green die is shown along the top margin of the following table.  Sums are shown in the interior of the table.  Outcomes belonging to event A are shown in red, outcomes belonging to event B are shown underlined, and those in event C are in italics.

        1 2 3 4 5 6
      1 2 3 4 5 6 7
      2 3 4 5 6 7 8
      3 4 5 6 7 8 9
      4 5 6 7 8 9 10
      5 6 7 8 9 10 11
      6 7 8 9 10 11 12

      • From the table shown above P[A]=(Number of red sums)/36=30/36, P[B]=(Number of underlined sums)/36=18/36, and P[C]=(Number of bold italicized sums)/36=5/36.  P[Not A]=1-P[A]=1-(30/36)=6/36, P[Not B]=1-P[B]=1-(18/36)=18/36, P[Not C]=1-P[C]=1-(5/36)=31/36.

      • From the table above P[A or B]=(Number of red or underlined (or both) sums)/36=33/36, P[A or C]=(Number of red or bold italicized sums)/36=31/36, and P[B or C]=(Number of underlined or bold italicized sums)/36=21/36.

      • From the table above P[A and B]=(Number of red and underlined sums)/36=15/36, P[A and C]=(Number of red and bold italicized sums)/36=4/36, and P[B and C]=(Number of underlined and bold italicized sums)/36=2/36.

         

    • Example 3: A card is drawn from an ordinary deck of 52 cards.  Events defined are: A=Black card, B=Spade,
      and C=Ace.  Find P[A], P[B], P[C], P[Not A], P[Not B], P[Not C], P[A or B],
      P[A or C], P[B or C], P[A and B], P[A and C], P[B and C].

      • The deck has 26 black cards, so P[A]=26/52, it has 13 spades, so P[B]=13/52, and it has 4 aces, so P[C]=4/52.

      • In the deck 13 cards are both black and spades, so P[A and B]=13/52, 2 cards are both black
        and aces, so P[A and C]=2/52, and one card is both a spade and an ace, so P[B and C]=1/52.

      • In an ordinary deck 26 cards are either black or a spade (or both) so P[A or B]=26/52, 16 cards are either black or an ace (or both), so P[A or C]=28/52, and 16 cards are either a spade or an ace (or both), so P[B or C]=16/52.

    • Example 4: A fair die is rolled and a fair coin is flipped the number of times shown on the die, e.g. if the die lands with a four facing up, the coin is flipped 4 times, if the die lands 1, the coin is flipped once, etc.  List a few sample points for this experiment.  How many sample points are there in the sample space of this experiment?  Should each sample point be assigned the same probability measure?

    • Example 5: A fair coin is tossed.  If this coin comes up heads, a fair coin is tossed, otherwise, a coin with probability of heads equal to 2/3 is tossed.  List all of the sample points in the experiment.  Should each sample point be assigned the same probability?

    • Example 6: Pick 2 balls without replacement from a container with 4 balls numbered 1 through 4.  Also,
      balls numbered 1 and 2 are green and balls numbered 3 and 4 are red.  List all of the sample points in the experiment.  What probability should be assigned to each of the sample points?

  • Conditional Probability and Independence

    • Rules for Conditional Probability

      • P[A|B] = P[A and B] / P[B]

      • or P[A and B] = P[A | B] P[B] or P[A and B] = P[B | A] P[A]
       
    • Conditional Probability Calculations

      • Consider drawing two cards, one after another without replacement, from an ordinary deck.  On drawing the first card the sample space consists of 52 cards.  However, once the first card is drawn, the sample space for the second draw consists of the 51 cards that remain.  Then, for example, the probability that the 2nd card is a heart given that the first card was a heart is 12/51, while the probability that the 2nd card is a heart given that the first card was not a heart is 13/51.  These probabilities are called conditional probabilities.  They are denoted by

        P[2nd is heart | 1st is heart] and P[2nd is heart | 1st is not heart].

        The vertical symbol | means given.

      • A ball is selected from a container that holds 6 red and 10 green balls.  Then a second ball is selected from the container without replacing the first ball.  The probability that the second ball is green given that the first ball is green is 9/15 while the probability that the second ball is green given that the first ball is red is 10/15.  These probabilities can be written as

        P[2nd ball green | 1st ball green]=9/15 and P[2nd ball green | 1st ball red]=10/15

      • A population is classified by gender and political party affiliation.  The membership of the population is shown in the following table with gender along the top and political party along the left side of the table.  The interior of the table givens the number of people in each classification.

          Female Male
        Democrat 45 50
        Republican 60 44
        Independent 10 15

        A person is selected at random from this population.  The following ordinary probabilities can be found directly from the table: P[Democrat]=95/224, P[Female]=115/224, P[Democrat and Female]=45/224.  Also, the following conditional probabilites can be found directly from the table by working with the appropriate row or column of the table.  P[Democrat | Female]= 45/115, P[Female | Democrat]= 45/95, and P[Independent | Male]= 15/109.  From this example,  notice that the following are true: 

        (1) P[Female | Democrat] = P[Female and Democrat]/P[Democrat]
        (2) P[Independent and Male] = P[Independent | Male] P[Male].

        The two rules, P[A | B] = P[A and B]/P[B] (for P[B] unequal to 0) and P[A and B] = P[A | B] P[B] are true for any two events, A and B.

    • Independence of Events: Events A and B are independent if and only if P[A | B] = P[A]

      • The definition of independence can be stated as A and B are independent if and only if the occurrence of one of them, say B, does not affect the occurrence of the other, A.

      • Since P[A and B] = P[A | B] P[B], and since for independent events, P[A | B]=P[A],
        P[A and B] = P[A] P[B]

      • In tossing a fair coin twice, the probability of event A, getting heads on the first toss is 1/2.  The probability of event B, getting heads on the second toss is also 1/2.  The probability of event A and B, getting heads on the first and second toss is 1/4.  In this case P[A and B] = P[A] P[B], so A and B are independent events.

      • In drawing two balls without replacement from a container that holds 6 red and 10 green balls.  The probability of event A, that the first ball picked is red, is 6/16.  The probability of event B, that the second ball picked is red, is also 6/16.  This second probability may seem surprising but it can be shown to be true.  The probability of event A and B, that both balls are red, is (6/16)(5/15).  In this case P[A and B] is not equal to P[A] P[B], that is A
        and B are not independent events.

      • If the engines on a jet airplane are identical in their ability to operate throughout a particular flight, if the probability that any one of the engines works properly for a particular flight is 0.97, if the engines operate independently of one another, and if the plane can make this particular flight safely if at least one engine continues working, what is the probability that a jet plane with four of these engines is able to make this flight safely?

        P[Makes flight safely] = 1 - P[Doesn't make flight safely] = 1 - P[all engines fail] = 1 - 0.034 = 0.99999919  The key step was finding the probability that all engines fail.  This was found by multiplying the probability that any one of the engines fail together 4 times--this is allowed by the assumption of independence for the engines.

  • Random Variables

    • Definition--A random variable is a quantitative variable whose value is determined by some chance mechanism.  Examples are the total number of heads in 10 tosses of a fair coin, the toss number of the first head in 10 tosses of a fair coin, the number of red balls selected when drawing 2 balls (without replacement) from a container that holds 8 red balls and 20 green balls.

    • Discrete and Continuous Random Variables

      • Discrete random variables can only have a finite or countably infinite number of values.  The number of heads in 10 tosses of a fair coin, the toss number of the first head if a fair coin is tossed until a head appears, or the number of green balls selected in the example given above.

      • Continuous random variables can assume any of an uncountably infinite set of values.  For example, if a point is picked at random on the interval from [0,1], there are an uncountably infinite number of values that could be picked.

    • Probability Distribution of a Discrete Random Variable--Each discrete random variable can only assume a finite or countably infinite number of values.  If a table is made associating the probability of each value with the value, the table or association is the probability distribution of that random variable.  As an example consider tossing a fair coin 3 times.  A random variable that can be associated with this experiment is a count of the number of heads in the 3 tosses.  Call this random variable T.  T can assume any of the values 0, 1, 2, or 3.  The probabilities associated with each of these values are P[T=0]=1/8, P[T=1]=3/8, P[T=2]=3/8, and P[T=3]=1/8.  These probabilities make up the probability distribution of this random variable.  In table form, this probability distribution is:

      t

      0

      1 2 3
      P[T=t] 1/8 3/8 3/8 1/8


      Notice that the sum of the probabilities is 1.  This is true for any discrete random variable. 

      Run the experiment of tossing a fair coin 3 times 1000 times updating after every 100 tosses.  To do this link here, and when the page opens click the red die in front of number 4.  Set the number of coins at 3.  After running the experiment 1000 times, what can you say about the theoretical (in blue) and actual (in red) probability distributions of the number of heads?

      In all examples of discrete random variables, the probabilities in the probability distribution table give the 'long-term' proportion of times that the random variable assumes each possible value.

    • Example 1: Find the probability distribution of the number of red balls selected if two balls are selected (without replacement) from a container which has 4 balls numbered 1 through 4, with balls numbered 1 and 2 red balls and balls numbered 3 and 4 green balls.

    • Example 2: Find the probability distribution of the number of red balls selected if two balls are selected (with replacement) from a container which has 4 balls numbered 1 through 4, with balls numbered 1 and 2 red
      balls and balls numbered 3 and 4 green balls.

    • Example 3: Find the probability distribution of the number of red balls selected if two balls are selected (with replacement) from a container with 20 balls, 10 of them red and 10 green.

    • Example 4: Find the probability distribution of the number of red balls selected if two balls are selected (without replacement) from a container with 20 balls, 10 of them red and 10 green.


      For examples 3 and 4, you can use this link to a simulation of the situation.  When the page opens click the red die in front of number 4 to open the simulation.  When the simulation opens, set N to 20, set R to 10, the number of red balls, and set n, the sample size to 2.  Select with or without replacement as appropriate for the example.  The blue graph and the text below it will show probabilities for each number of red balls.

    • Example 5: In examples 3 and 4 decrease the number of red balls.  What happens to the probability distribution of the number of red balls in the sample?  Is this expected?  Also, for each number of red balls, observe the differences in the probability distribution with and without replacement.  Next, set the number of red balls at 10, use sampling with or without replacement, and run the simulation of drawing 2 balls, 1000 times, updating every 100 times.  What do you see?


    • Mean (also called Expected Value) and Standard Deviation of a Discrete Random Variable

      Consider again the count of heads in 3 tosses of a fair coin.  If this experiment is repeated, say 10 times, and the
      number of heads in each series of 3 tosses is counted, you will have a set of numbers like 0,1,3,1,2,2,1,1,3,0.  The average of these numbers is the average value for the random variable, the number of heads in 3 tosses of a fair coin.  To see what happens in a larger number of runs of the experiment, again link here, and when the page opens click the red die in front of number 4.  Set the number of coins at 3 and run the experiment 100 times.  What is the average number of heads per 3 tosses?  Now reset and run the experiment 1000 times.  What is the average number of heads per 3 tosses?

      The long-term average number of heads is called the expected value of the random variable, the number of heads in 3 tosses of a fair coin.  This expected value can be found for most random variables.  Think of expected value as the average value of a random variable.

      There is an easier way to find the expected value of this (or any) discrete random variable.  If the experiment of tossing the coin 3 times is repeated for a large number, N, times, the experiment will end in 0 heads n0 times, in 1 head n1 times, in 2 heads n2 times, and in 3 heads n3 times.  The total number of heads is 0 n0 + 1 n1 + 2 n2 + 3 n3, and the average number of heads per run of the experiment is

      (0 n0 + 1 n1 + 2 n2 + 3 n3)/N = 0 (n0/N) + 1 (n1/N) + 2 (n2/N) + 3 (n3/N)

      For large N, (n0/N) ~ P[0 Heads], (n1/N) ~ P[1 Head], (n2/N) ~ P[2 Heads], (n3/N) ~ P[3 Heads], so the average number of heads per run of the experiment is

      0 P[0 Heads] + 1P[1 Head] + 2 P[2 Heads] + 3 P[3 Heads]

      This is called the Expected Value or Mean and is denoted, for a general random variable X, by E[X].  It can be computed by

      Using this formula on the random variable T, the total number of heads in 3 tosses of a fair coin, you get

      =E[T] = 0 (1/8) + 1 (3/8) + 2 (3/8) + 3 (1/8) = 12/8 = 3/2 = 1.5.  This can be interpreted as the average number of heads per sequence of 3 tosses if the experiment is repeated a large number of times.

      Just as you are able to find the average value for a random variable, so you can also find the standard deviation of the random variable.  In the case of a random variable, the standard deviation is given by

      For random variable T, the total number of heads in 3 tosses of a fair coin, the standard deviation computed by the rightmost formula is

      SD[T] = Square Root of [02 (1/8) + 12 (3/8) + 22 (3/8) + 32 (1/8) - (3/2)2] = Square Root of [3/4] = 31/2/2


  • The Binomial Random Variable

    • Definition--A binomial random variable with parameters n and p is a count of the number of successes in n experiments (or trials).

      • Each trial can result in only two outcomes, a success, S, or a failure, F.
      • The probability of a success on any trial is P[S] = p and the probability of a failure on any trial is P[F] = 1-p = q
      • The outcome of any trial has no effect on outcomes of other trials.
      • The binomial random variable is a count of the number of successes in n trials.

      An example is the toss of a fair coin 3 times.  If you think of a success as a head, the count of the number of heads in 3 tosses satisfies the definition of a binomial random variable.  Here n=3 and p=1/2.

      Another example is the drawing of 2 balls with replacement from a container with 20 balls, 10 of which are red and 10 white.   A red ball is considered a success.  Then the count of red balls is a binomial random variable with n=2 and p=1/2.  If the drawing is without replacement, the probabilities of a red ball change from draw to draw, so the count of red balls is no longer a binomial random variable.  However, when the number of balls drawn is much smaller than the total number of balls in the container, the count of red balls will be distributed almost like a binomial random variable.

    • Probability Distribution of a Binomial Random Variable with parameters n and p.

      Since the binomial random variable is a count of the number of successes in n trials, the number of successes can only be an integer between 0 and n.  Thus, if X is the total number of successes in n trials, P[X=k] will only be nonzero for k=0,1,2,3,...,n.  The formula for P[X=k] is





    where




    • Mean and Standard Deviation for a binomial random variable with parameters n and p.

      Formulas for the mean or expected value and standard deviation of any random variable have been given above.  When these formulas are applied to a binomial random variable with parameters n and p, they simplify to
      the following expressions:



           and  

    • Examples

      • According to information collected in the last census, 20% of working women have never been married.  If 10 working women are selected at random, what is the probability that exactly 3 of them have never been married?  What is the probability that at most 2 of them have never been married?  What is the probability that all 10 of them have never been married?  How many would you expect to be never married women?

      • If a person makes random guesses at each of the questions on a 20 question true-false test, what is the probability that they will get exactly 10 questions correct?  What is the probability that they will get none of the questions correct?  What is the probability that they will get at least 1 correct answer?  How many questions would you expect them to get correct?

      • You draw 5 balls (with replacement) from a container in which 30 of the balls are red and 20 are green.  What is the probability that none of the balls that you pick are green?  What is the probability that at least one of the balls selected is green?  What are the mean and standard deviation of the number of red balls selected?

      • You play a game in which you toss a pair of fair dice and win a prize of $5 if the sum of numbers on the dice is 6.  You play the game 12 times.  What is the probability that you win $5 in exactly 3 of those games?  In no game?  In at least one game?  How many times would you expect to win?