binomial prob -advanced?
<< The Idiot awards? I THINK SOMEONES IN MY HOUSE HELP URGENT? >>
Mail this post
Sphere: Related Content
A home security system is designed to be triggered in 96% of attempted burglaries. If 10 homes equipped with such a system experience an attempted burglary, find the probability that at most 7 alarms are triggered. Round your answer to four decimal places.
Mail this post
Sphere: Related Content
Articulos relacionados
- No related posts
P(at most 7)
= 1 – P(8) – P(9) – P(10)
= 1 – 10C8*.96^8*.04^2 – 10C9*.96^9*.04^1 – 10C10*.96^10*.04^0
= 1 – .0519400497 – .2770135983 – .664832636
= .006213716
= .0062
There is a 0.62% chance that at most 7 alarms are triggered when 10 homes equipped with the security system experience an attempted burglary.
November 4th, 2009 at 12:00 amLet X be the number of alarms triggered. X has the binomial distribution with n = 10 trials and success probability p = 0.96
In general, if X has the binomial distribution with n trials and a success probability of p then
P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)
for values of x = 0, 1, 2, …, n
P[X = x] = 0 for any other value of x.
The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n – x failures.
Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.
X ~ Binomial( n , p )
the mean of the binomial distribution is n * p = 9.6
the variance of the binomial distribution is n * p * (1 – p) = 0.384
the standard deviation is the square root of the variance = √ ( n * p * (1 – p)) = 0.6196773
The Probability Mass Function, PMF,
f(X) = P(X = x) is:
P( X = 0 ) = 1.048576e-14
P( X = 1 ) = 2.516582e-12
P( X = 2 ) = 2.717909e-10
P( X = 3 ) = 1.739462e-08
P( X = 4 ) = 7.30574e-07
P( X = 5 ) = 2.104053e-05
P( X = 6 ) = 0.0004208106
P( X = 7 ) = 0.005771117
P( X = 8 ) = 0.05194005
P( X = 9 ) = 0.2770136
P( X = 10 ) = 0.6648326
The Cumulative Distribution Function, CDF,
F(X) = P(X ≤ x) is:
x
∑ P(X = t) =
t = 0
P( X ≤ 0 ) = 1.048576e-14
November 4th, 2009 at 12:00 amP( X ≤ 1 ) = 2.527068e-12
P( X ≤ 2 ) = 2.743180e-10
P( X ≤ 3 ) = 1.766894e-08
P( X ≤ 4 ) = 7.482429e-07
P( X ≤ 5 ) = 2.178877e-05
P( X ≤ 6 ) = 0.0004425994
P( X ≤ 7 ) = 0.006213716 ← answer
P( X ≤ 8 ) = 0.05815377
P( X ≤ 9 ) = 0.3351674
P( X ≤ 10 ) = 1