The rate of defects among cd players of a certain brand is 1.5%. use the poisson approximation to 19) the binomial distribution to find the probability that among 430 such cd players received by a store, there are exactly three defective cd players.
The poison distribution can be used to approximate the binomial distribution when the sample size n is large. This is then calculated using the formula
P(X) = e^-(np) *(np)^x
substituting X= 3
P(3)= e^-(430*1.5/100) * (430*1.5/100)^3
P(3)= 0.00158*268.3
P(3) = 0.42
P(3) = 0.0042%
P= probability of X occurring given n and p
n= sample size
p= true probability
e= exponential constant ~2.718
X=number of sample successes
The solution for the problem using binomial distribution: Given: p = 1.5% or 0.015 n = 430 Mean = lambda = m = np = 430*0.015 = 6.45 According to Poisson distribution: P(x) = e^-(np) *(np) ^x Where: P is the probability of x n is the sample size e is the exponential constant P(3)= e^-(430*1.5/100) * (430*1.5/100)^3 P(3)= 0.00158* 268.3336125 P(3) = 0.4240%
