The fingerprint of every finger can be classified into one of three basic patterns: a loop, a  whorl, or an arch. Forensic experts have determined that about 65% of all fingers have loops, 30% have  whorls, and 5% have arches. Suppose that these percentages are exactly correct and patterns are independent  from one finger to the next. Select a person at random and classify each of their fingerprints on both hands.  Let X = the number of fingerprints with a loop.  (a) Calculate and interpret the mean of X.  (b) Calculate and interpret the standard deviation of X.

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 The fingerprint of every finger can be classified into one of three basic patterns: a loop, a  whorl, or an arch. Forensic experts have determined that about 65% of all fingers have loops, 30% have  whorls, and 5% have arches. Suppose that these percentages are exactly correct and patterns are independent  from one finger to the next. Select a person at random and classify each of their fingerprints on both hands.  Let X = the number of fingerprints with a loop. 

(a) Calculate and interpret the mean of X

(b) Calculate and interpret the standard deviation of X

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Step 1

Binomial Distribution : 

A binomial experiment is a statistical experiment that has the following properties:

  • The experiment consists of n repeated trials.
  • Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
  • The probability of success, denoted by P, is the same on every trial.
  • The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials. 

Notation : 

The following notation is helpful, when we talk about binomial probability.

  • x: The number of successes that result from the binomial experiment.
  • n: The number of trials in the binomial experiment. 
  • P: The probability of success on an individual trial.
  • nCr: The number of combinations of n things, taken r at a time. 

A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution.  

The binomial distribution has the following properties:

  • The mean of the distribution (μx) is equal to n * P .
  • The variance (σ2x) is n * P * ( 1 - P ).
  • The standard deviation (σx) is sqrt[ n * P * ( 1 - P ) ].

 

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