Question

A box in a machine shop consists of 5 coated and 10 uncoated cutting inserts which are of otherwise similar characteristics. The inserts got mixed up randomly in the box. An operator has taken 4 of them at once without noticing the differences to mount on a 4-tooth face milling cutter that uses inserts.

The probability of the milling cutter having all uncoated inserts is ______ (rounded off to two decimal places).

Hint:

Use the multiplication rule for drawing without replacement: \( \frac{10}{15} \times \frac{9}{14} \times \frac{8}{13} \times \frac{7}{12} \), or equivalently the ratio of combinations \( \frac{^{10}C_4}{^{15}C_4} \).

Answer 1

Correct Answer: 0.15

Step 1: Understanding the Concept:
There are 15 inserts in the box, 5 coated and 10 uncoated, and 4 of them are picked together without noticing which is which.
This is a selection problem, so we use combinations rather than ordered probability.
We want the chance that the group of 4 chosen inserts is made up entirely of uncoated pieces.

Step 2: Key Formula or Approach:
When items are chosen "at once" (unordered, without replacement), the probability is the ratio of favourable combinations to total combinations.
\[ P = \frac{{}^{10}C_4}{{}^{15}C_4} \]

Step 3: Detailed Explanation:
Total ways to choose any 4 inserts out of 15:
\[ {}^{15}C_4 = \frac{15 \times 14 \times 13 \times 12}{4 \times 3 \times 2 \times 1} = 1365 \]
Ways to choose 4 uncoated inserts out of the 10 available uncoated ones:
\[ {}^{10}C_4 = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]
So the probability is:
\[ P = \frac{210}{1365} = \frac{2}{13} = 0.1538 \]

Final Answer:
Rounding this to two decimal places gives the required probability. \[ \boxed{0.15} \]