Probability Calculation for Uncoated Inserts

This problem involves calculating the probability of a specific outcome when selecting items randomly from a group. We need to find the chance that all 4 selected cutting inserts are uncoated.

Understanding the Problem

• Total number of cutting inserts = 5 (coated) + 10 (uncoated) = 15.
• Number of inserts selected = 4.
• We need the probability that all 4 selected inserts are from the 10 uncoated ones.

Step 1: Calculate Total Possible Selections

The total number of ways to choose any 4 inserts from the 15 available is calculated using combinations ($C(n, k)$), where $n$ is the total number of items and $k$ is the number of items to choose.

Total combinations = $C(15, 4) = \frac{15!}{4!(15-4)!} = \frac{15!}{4!11!}$

Calculation: $\frac{15 \times 14 \times 13 \times 12}{4 \times 3 \times 2 \times 1} = 1365$ ways.

Step 2: Calculate Favorable Selections

The number of ways to choose 4 uncoated inserts from the 10 available uncoated inserts is also calculated using combinations.

Favorable combinations = $C(10, 4) = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!}$

Calculation: $\frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210$ ways.

Step 3: Determine the Probability

The probability is the ratio of favorable selections to the total possible selections.

Probability (all 4 uncoated) = $\frac{\text{Favorable combinations}}{\text{Total combinations}} = \frac{C(10, 4)}{C(15, 4)}$

Probability = $\frac{210}{1365}$

Step 4: Final Calculation and Rounding

Simplify the fraction and convert to a decimal:

Probability = $\frac{210}{1365} = \frac{2}{13} \approx 0.1538...$

Rounding to two decimal places, the probability is 0.15.