The probability of the milling cutter having all uncoated inserts is ______ (rounded off to two decimal places).
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.
Important Questions from Probability
- If one event is unaffected by the outcome of another event, the two events are said to be:
1. Dependent
2. Independent
3. Mutually exclusive
4. Mutually inclusive - Let $M$ be a randomly chosen non-empty subset of $S = \{1, 2, 3, \dots, 2026\}$.
Which of the following is the probability that the product of all the elements of $M$ is even? - A fair coin is tossed 6 times. The probability of getting at least 5 heads is _________________. (rounded off to two decimal places)
- Consider a random experiment of rolling a fair die. The probability of getting an odd number or a number less than 4 is ______ (Rounded off to two decimal places).
- In a null hypothesis significance test, the significance level $\alpha$ was set to 0.01. For which one or more of the following $p$ values would the null hypothesis be rejected?
