Answered

Shelia does quality control for a company that manufactures lawn mower parts. On any given day, she finds the probability distribution for defective parts as shown in the table.





\begin{tabular}[t]{l}
Defective Parts:

$X$
&


Probability:

$P(X)$
\\

\hline 0 & 0.85 \\

\hline 1 & 0.10 \\

\hline 2 & 0.04 \\

\hline 3 & 0.01 \\

\hline

\end{tabular}



Using the data from the table, what is the probability of having 2 defective parts in a day?

Verified Answer :

  • Analyze the probability distribution table for the random variable \(X\), representing the number of defective parts.
  • Locate the row where the number of defective parts \(X = 2\).
  • Identify the corresponding probability \(P(X=2) = 0.04\) from the table.
  • The probability of having exactly 2 defective parts is \(\boxed{0.04}\).

Explanation

  1. Understand the problem and the provided data

We are given a probability distribution table representing the number of defective lawn mower parts found by Shelia in a single day. The random variable \(X\) represents the number of defective parts, and \(P(X)\) represents the probability of that event occurring.

The distribution is:

  • \(P(X=0) = 0.85\)
  • \(P(X=1) = 0.10\)
  • \(P(X=2) = 0.04\)
  • \(P(X=3) = 0.01\)

The objective is to find the probability that there are exactly \(2\) defective parts in a day, which is represented by \(P(X=2)\).

  1. Identify the requested probability from the table

Looking at the table provided:

  • For \(X = 0\), \(P(X) = 0.85\)
  • For \(X = 1\), \(P(X) = 0.10\)
  • For \(X = 2\), \(P(X) = 0.04\)
  • For \(X = 3\), \(P(X) = 0.01\)

The value we are interested in is the probability corresponding to \(X = 2\). By observing the table, the value associated with \(X = 2\) is \(0.04\).

  1. State the final result

Since the probability distribution is already given in the table, we simply extract the value corresponding to \(X = 2\). Thus, the probability of having exactly \(2\) defective parts in a day is \(0.04\).

Examples

In quality control, probability distributions are essential for predicting outcomes and managing risks. For instance, if a manager knows there's a \(4\%\) probability of having exactly \(2\) defective parts, they can plan their daily resource allocation and supply chain logistics more effectively, ensuring that they have enough replacement parts on hand without overstocking. This helps in maintaining high operational efficiency and minimizing costs.

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