Binomial Probabilities

Probabilities are numbers between 0 and 1, inclusive, that indicate the likelihood of some event or outcome occurring. If an experiment is performed, the resulting basic outcomes comprise a sample space for the experiment.  The sum of the probabilities associated with each of the basic elements in the sample space is always equal to 1.  For example, if an experiment involves tossing a fair coin twice and observing how they fall, a sample space for the experiment is S = {HT, HH, TH, TT}.  If the coin is fair, each of the four outcomes have a probability of 1/4 of occurring and the sum of the four probabilities is 1/4 + 1/4 + 1/4 + 1/4 = 1.  We summarize this discussion below.

Basic Properties of Probability
  • Probability is a number between 0 and 1.
  • The sum of the basic probabilities associated with the sample space of an experiment is 1.

The study of experiments typically involves calculating probabilities associated with some sample space of the experiment. 

 


Exploration I

The purpose of this exploration is to introduce binomial experiments.

Consider the experiment of tossing a fair coin three times and observing the number of heads that result. This experiment possesses the following characteristics:

  •  It consists of three identical trials of tossing one coin.

  •  Each trial results in exactly one of two outcomes, heads or tails.

  •  The trials are independent, that is, the outcome of one trial does not affect the outcome of any other.

  •  The probability of getting a head remains constant from trial to trial, the probability being   0.5.

1. There are eight different equally likely possibilities that can result from tossing a coin three times. List      them. Hint: HHT is one of the possibilities.

2. If the coin is fair, each of the eight outcomes is equally likely to occur. What is the the probability     assigned to each outcome? Remember that probability is the likelihood that an outcome occurs.

3. Suppose we are interested in the number of heads, X, that occur in three tosses of the coin. How many     values can X take on. Since there are a finite number of values, the variable X is called a discrete random     variable.

4. For each value of X, compute the probability of X, denoted by P(X).

       X    P(X)
   
   
   
   

 

5. Show that the sum of the probabilities is 1.

6. Construct a vertical line-segment graph with the probabilities assigned to the vertical axis and the values     of  X assigned to the horizontal axis. The line-segment graph is symmetric about what value?

 

 

Exploration II

The experiment described in Exploration 1 is an example of a class of experiments, called binomial experiments. In general, a binomial experiment is an experiment which possesses the following four properties:

Characteristics of a Binomial Experiment

  • The experiment consists of n identical trials.

  • Each trial results in exactly one of two outcomes, called success or failure. A success is denoted by S and a failure is denoted by F.

  • The n trials are independent.

  • The probability of success, p, remains constant from trial to trial.

The probability distribution for the number of successes is called a binomial distribution .

1. Which of the following situations can be modeled using a binomial experiment? Explain.

  a. Administering a cold medication with a success rate of 0.90 to ten people with colds and observing the       number of cures  [binomial]

  b. Tossing a thumbtack eight times and observing the number of times it lands point up

  c. Guessing on a ten-question true-false test and observing the number of correct answers

  d. Tossing a die six times and observing the numbers of 3's that result  [binomial]

  e Observing a baseball player with a 0.400 batting average and counting the number of base hits he gets      [not binomial]

  f. Forty percent of all campers at a certain summer camp contract poison ivy. Eight students attend and we      are interested in the number who contract poison ivy.

  g. A survey of the residents in a certain town, indicated that 30% of the residents favor building a community       center and 70% are opposed. Ten residents are randomly surveyed and asked if they favor the proposed       new community center. [binomial]

  h. A six-sided die is tossed five times and the sum of the faces showing is to be determined.

  i. Out of the next ten babies born at Memorial Hospital, the number of males is to be determined.       (Assume that male and female births are equally likely.)   [binomial]

  j. At a certain college, 40% of entering freshmen eventually graduate. Of 30 freshmen who enroll next      semester, the number who eventually graduate is to be determined.

  k. Four candidates are running for governor. A survey is conducted to determine voters' support for the four       candidates. [not binomial]

 

 

 

Exploration III

This exploration introduces the notation used for binomial experiments. The following symbols are used to describe binomial experiments:

  •  S = success

  •  F = failure

  •  p = P(S)

  •  1 - p = P(F)

  •  n = number of trials

  •  X = number of successes
        

The values n and p are called parameters, which completely determine a binomial probability distribution.

 

1. A cross fertilization of related species of white-flowered and blue-flowered plants produces offspring of     which 20% are white-flowered plants. Six blue- flowered plants were paired and crossed with six white     flowered plants, and it was found that there were two white-flowered plants among the six offspring. Is this     a binomial experiment? If so, identify a trial, a success, and the values for p , n, and X and P(F).

2. Mary tossed a six-sided die ten times to determine the number of 1s resulting. She obtained three 1s. Is     this a binomial experiment? If so, what constitutes a trial, a success, and a failure? What are the values of     n, p, and X? [A trial is tossing a die and observing the outcome , S  is observing a 1, and F is observing a     number different from 1.  n = 10, p = 1/6, X = 3, and P(F) = 5/6]

Success is the term used to describe the outcome of interest for a binomial experiment. It does not necessarily correspond to a "good" event.

3. Suppose a certain type of medication will not cause a skin reaction in 90% of the people who use it. We     are interested in the number of people out of the next five who use it and have a skin reaction. Identify a     trial,  a success, the values for p and n, and the possible values of X.

 

Probabilities associated with a binomial experiment having a large sample space are typically calculated using the binomial probability formula.  

Binomial Probability Formula

P(x) = C(n, x) px (1 - p)n - x


where C(n, x) is the binomial coefficient defined by

Binomial Coefficient

C(n, x) = n! / [x!( n - x)!]



The symbol n! is called n factorial and is defined by

n factorial

n! = n (n - 1)(n - 2) ... (1) 


For example, 4! = (4)(3)(2)(1) = 24.
By definition 0! = 1.

 

Exploration IV

The purpose of this exploration is to use the binomial probability formula to calculate probabilities.  For problems 1 - 4, return to the binomial experiment in Exploration 1 above.

1.  Calculate the binomial coefficient C(3, 0), C(3, 1), C(3, 2) and C(3, 3) by using the binomial coefficient      formula.

2.  Calculate the binomial probabilities P(0), P(1), P(2), and P(3) by using the binomial probability formula.

3.  Check the probabilities in (2) by comparing them to those obtained in Exploration I.  

4.  Explain why C(n, x) is called a binomial coefficient.

5.  If an experiment consists of tossing a fair coin four times and observing the results, how many elements      does the sample space of equally likely outcomes have? List them.

6. If the variable X denotes the number of heads obtained in (5), determine the probabilities without using the     binomial probability formula.

      X     P(X)
      0  
      1  
      2  
      3  
      4  

 

7.  Use the binomial probability formula to calculate the 5 probabilities in (6). Check your results by      comparing your probabilities with those obtained in (6).

 


Exploration V

The purpose of this exploration is to become proficient using the online binomial calculator. In the problems that follow, X is assumed to be a binomial random variable.

1. For n = 6 and p = 0.5, find the values of P(X = r) for r = 0, 1, 2, 3, 4, 5, and 6.

2. Determine the sum of the seven probabilities. [1]

3. Find the sum of the probabilities associated with the following binomial parameters.

   a. n = 5 and p = 0.4

   b. n = 4 and p = 0.6

   c. n = 5 and p = 0.3

4. Form a generalization concerning the sum of the binomial probabilities associated with parameters n and     p. [The sum of the probabilities always equals 1]

5. Let n = 20 and p = 0.55. Determine each of the following probabilities using the binomial calculators in two     different ways.

  a. P( X greater than or equal to 18 ).   [one way is P(18) + P(19) + P(20); another is 1 - P(X < 18)  The       answer is 0.0009.]  Note that the notation 8.0E-4 is scientific notation meaning 0.0008.  


  b. P( 9 < X < 13 ).


  c. P( X > 12 ).


  d. P( X less than or equal to 3 ).


  e. P( X < 8 or X > 14 ). [One way is 1 - P(8) - P(9) - P(10) - P(11) - P(12) - P(13) - P(14).  Why?                 Another way is P(X < 8) +  (1 - P(X < 15)) which is equivalent to 1 + P(X < 8) - P(X < 15).  Why?  The       answer is 0.1133.]


  f. P( 10 < X < 13 ).


  g. P( 0 < X < 20 ). 


  h. P( 0 < X < 21 ).
  [The answer is not equal to 1, but to four decimal places is rounded to 1]

 

Exploration VI

For each of the following exercises, you should verify that the exercise involves a binomial experiment and identify the values of p, n, and X before solving the exercise.

1. A recent survey showed that 60% of college students smoke. What is the probability that of seven     students surveyed,

  a. three of them smoke?  [n = 7, p = 0.60, X = 3; P(3) = 0.1935] 


  b. none of them smoke?


  c. at most two of them smoke? [P(0) + P(1) + P(2) = 0.0963]


  d. three or four of them smoke?


  e. they all smoke?


  f. less than 5 smoke?

 

2. If a baseball player with a batting average of 0.600 comes to bat five times in a game, what is the     probability he will get

  a. three hits?

  b. no hits?

  c. five hits?

  d. six hits?

  e. at least 2 hits?   [P(X > 1) = 1 - P(X <  2) = 0.913. Why?]

  f. at most 3 hits?

 

3. It was found that 40% of the campers at a certain summer camp contacted poison ivy. If eight students     attend camp this summer, find the probability that:

  a. all will contract poison ivy.

  b. two will contract poison ivy.

  c. at most three will contract poison ivy.  

  d. at least seven will contract poison ivy.

 

4. A survey of the residents in a certain town showed that 30% favored brand W toothpaste. If ten of the     residents are randomly surveyed at a grocery store, what is the probability that:

  a. no one favors brand W?  

  b. four don't favor brand W?   [ n = 10, p = 0.7, X = 4; P(4) = 0.0368 or n = 10, p = 0.3, X = 6; P(6) =       0.0368.]

  c. at least eight favor brand W?

  d. at most two favor brand W?

  e. all favor brand W?

 

5. Dick has been observed to make 65% of his free-throw shots during basketball games. What is the     probability that Dick will make:

  a. three of the next six shots?

  b. five of the next ten shots?

  c. all of the next four shots?

 

6. In a certain city, 40% of the registered voters are Democrats. If nine voters are randomly selected, find the     probability that:

  a. two of them are Democrats.

  b. at least one of them is a Democrat.

  c. at least eight are Democrats.

  d. at most three are Democrats.