A widget company produces 25 widgets a day, 5 of which are defective. Find the probability of selecting 5 widgets from the 25 produced where none are defective.
A. [tex]\frac{1}{3125}[/tex] B. [tex]\frac{1}{5}[/tex] C. [tex]\frac{2584}{8855}[/tex] D. [tex]\frac{1}{10626}[/tex]
To find the probability of selecting 5 widgets from the 25 produced, where none are defective, we need to approach this problem step by step.
1. Total Number of Widgets: The company produces 25 widgets each day.
2. Defective Widgets: Out of these 25 widgets, 5 are defective.
3. Selecting Widgets: We need to calculate the probability of selecting 5 widgets where none are defective.
Step 1: Calculate the Total Number of Ways to Select 5 Widgets from 25
To find the total number of ways to select 5 widgets from 25, we use the combination formula: [tex]\[
\binom{25}{5} = \frac{25!}{5!(25-5)!}
\][/tex]
According to our result: [tex]\[
\binom{25}{5} = 53130
\][/tex]
Step 2: Calculate the Number of Ways to Select 5 Non-defective Widgets
Since there are 20 non-defective widgets (25 total widgets minus 5 defective widgets), we use the combination formula to find the number of ways to select 5 non-defective widgets from these 20: [tex]\[
\binom{20}{5} = \frac{20!}{5!(20-5)!}
\][/tex]
According to our result: [tex]\[
\binom{20}{5} = 15504
\][/tex]
Step 3: Calculate the Probability
The probability of selecting 5 widgets and finding none defective is the ratio of the number of ways to select 5 non-defective widgets to the total number of ways to select 5 widgets from the 25: [tex]\[
\text{Probability} = \frac{\binom{20}{5}}{\binom{25}{5}} = \frac{15504}{53130}
\][/tex]
Step 4: Simplify the Fraction
When simplified, the fraction [tex]\(\frac{15504}{53130}\)[/tex] reduces to: [tex]\[
\frac{2584}{8855}
\][/tex]
Step 5: Conclusion
Thus, the probability of selecting 5 widgets from the 25 produced where none are defective is:
[tex]\[
\boxed{\frac{2584}{8855}}
\][/tex]
Therefore, the correct answer is C. [tex]\(\frac{2584}{8855}\)[/tex].
