In: Statistics and Probability
A research team conducted a study showing that approximately 20% of all businessmen who wear ties wear them so tightly that they actually reduce blood flow to the brain, diminishing cerebral functions. At a board meeting of 15 businessmen, all of whom wear ties, what are the following probabilities? (Round your answers to three decimal places.) (a) at least one tie is too tight (b) more than two ties are too tight (c) no tie is too tight (d) at least 13 ties are not too tight
Here we have n = 15, p = 0.20
There two outputs to the experiment that wear tie too tight and wear no tie too tight.
So , x ~ B ( n , p )
p( X = x ) = nCx * px * ( 1-p )n-x
a) p ( x 1 )
= 1 - p ( x < 1 )
= 1- p ( x = 0 )
= 1 - 15C0 * 0.200 * ( 1-0.20 )15-0
= 1 - 0.0352
= 0.9648
b) p ( x > 2 )
= 1 - p ( x 2 )
= 1 - { p ( x = 0 ) + p ( x = 1 ) + p ( x = 2 ) }
= 1 - { 15C0 * 0.200 * ( 1-0.20 )15-0 + 15C1 * 0.201 * ( 1-0.20 )15-1 + 15C2 * 0.202 * ( 1-0.20 )15-2 }
= 1 - { 0.0352 + 0.1319 + 0.2309 }
= 1 - 0.3980
= 0.6020
c) p ( x = 0 ) = 15C0 * 0.200 * ( 1-0.20 )15-0 = 0.0352
d) probability that tie is too tight is 0.20. Probability that tie is not too tight 0.8 . So p = 0.8
Here we need to find that at least 13 ties are not too tight.
p ( x 13 ) = p ( x = 13 ) + p ( x = 14 ) + p ( x = 15 )
=15C14*0.8013 *(1-0.80 )15-13 +15C14* 0.8014 *(1-0.80 )15-14+15C15* 0.8015* (1-0.20)15-15
=0.2309 + 0.1319 + 0.0352
= 0.3980