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.
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.
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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