Question

you are the class president and have ordered 300 t - shirts for the grade from a clothing manufacturer that claims that it averages only five discolored t - shirts for every 50,000 t - shirts it produces. what is the probability that your order will contain one or more discolored t - shirts?
____% (round your answer to the nearest hundredth)

Explanation:

Step1: Calculate the probability of a single T - shirt being discolored

The probability \(p\) of a T - shirt being discolored is \(p=\frac{5}{50000}= 0.0001\)

Step2: Use the Poisson approximation

Since \(n = 300\) (number of trials) and \(p=0.0001\), and \(\lambda=np\). So \(\lambda=300\times0.0001 = 0.03\)
The probability mass function of a Poisson distribution is \(P(X = k)=\frac{e^{-\lambda}\lambda^{k}}{k!}\), where \(X\) is the random variable representing the number of discolored T - shirts.
The probability of having \(0\) discolored T - shirts is \(P(X = 0)=\frac{e^{- 0.03}(0.03)^{0}}{0!}=e^{-0.03}\)

Step3: Calculate the probability of having one or more discolored T - shirts

The probability of having one or more discolored T - shirts is \(P(X\geq1)=1 - P(X = 0)\)
Since \(P(X = 0)=e^{-0.03}\approx0.970446\), then \(P(X\geq1)=1 - 0.970446 = 0.029554\)

Answer:

\(2.96\%\)