Question

A worn out bottling machine does not properly apply caps to 5% of the bottles it fills. If you randomly select 20 bottles from those produced by this machine, what is the approximate probability that between 2 and 6 (inclusive) caps have been improperly applied?.

Answer 1

Probability would been equal to 0.9841 if we randomly select 20 bottles it fills.

What is Probability ?

The ratio of good outcomes to all possible outcomes of an event is known as the probability. The number of the  positive results for the experiment with 'n' outcomes can also be represented by the symbol x. The probability of an event can be calculated using the following formula.

Probability(Event) = Positive Results/Total Results = x/n

Given,

n = 20

p = 0.05

Thia information gives information about binomial distribution with parameter n = 20 , p = 0.05

Let X be the number of bottles have caps that she not properly applied

    X∝ B( n=20 , p= 0.05)

The p.m.f of X is

P( X=0)= ($\frac{20}{x}$) ( 0.05)ˣ (1 - 0.05)²⁰ ⁻ˣ

                                                       ; x = 0 , 1, 2.......20

                                                         0 < p < 1

                                                         q= 1 - p

= 0                                                    ; otherwise

P {x ≤ 3} = P { X = 0 } + P{ x= 1 } + P { x= 2 } + P { x= 3}

P( X=0)= ($\frac{20}{0}$) ( 0.05)⁰ (1 - 0.05)²⁰ ⁻⁰  = 0.3585

P( X = 1)= 0.3774

P( X= 2) = 0.1887

P( X= 3) = 0.0596

                = 0.3585+ 0.3774 +  0.1887+ 0.0596

                 = 0.9841

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