Binomial Probability

• If I tossed a coin three times, what is the probability that a heads is seen exactly two times.
  • We could build a tree diagram, and count all combinations (3) and divide by 8.
  • We could do some fancy thought experiment and come up with 3/8
  • We could do a sum of and computations

      P(first head, second head, third tail) 
    + P(first head, second tail, third head)
    + P(first tail, second head, third head) = 3/8
    	
     But if you notice, there are 3C2  different ways we can get two heads and a tail.
    	
     And the probability of a head is 1/2, so (1/2)2 × (1/2) is the probability of getting this combination.
    	
     So we could do 3C2 (1/2)2(1/2)
    	

• This type of problem is different from all the others we have been doing
  • With the exception of possibly expected value.
  • In this case we know the probability of an event.
  • And we want to predict if that will occur a number of times.
  • So like expected value, we are doing the experiment over and over.
  • But we are not concerned with profit/loss, just success
• The binomial probability formula
  • Is used when there are n independent trials of the same experiment
  • Each trial has two possible outcomes: success or failure
  • The probability of success or failure remains the same.
• The formula:
  • P(x) = _nC_x p^x(1-p)^n-x
• An egg distributor determines that the probability that an individual egg has a crack is 14%. In a box of one dozen eggs, what is the probability that exactly two eggs are cracked?

  The experiment is look at an egg.
  The event is that an egg is cracked
  
  P(E) = .14
  
  We will do this experiment 12 times and record success or failure for each time.
  We are looking to see the probability that the experiment succeeds twice.
  
  P(2) = 12C2(.14)2(.86)10
      

• This is an example of a probability distribution.
  • A probability distribution assigns a probability to each subset of the outcomes of an experiment.
  • In this case, 0,1,2, ... 12 eggs are cracked.
  • 
  • The sum of all probabilities is 1 (as it should be)
  • But this describes all experiments of this type.
  • This saves us the complex computations we would otherwise have to make.
• Some problems
  • According to a survey conducted by American Express 77% of adults stated they wan access to their cell phones and laptops while on vacation. If ten adults are sampled at random, determine the probability that exactly six want access to cell phones on vacation.
  • Records from a bank show that 70% of car loan applications are approved. If eight car loan applications are selected at random, determine the probability that exactly 8 are approved.
  • A quality control engineer at a GE lightbulb plant finds that 1% of all bulbs are defective. Determine the probability that exactly two of the next six bulbs made are defective.