Question

Silk scarves are produced in a factory. Quality control investigations find that 0.5% of the scarves produced have flaws, which reduce their value. A sample of 30 scarves is selected from the factory. Calculate the probability that the sample has:Exactly one flawed scarfNo flawed scarvesMore than three flawed scarves

Answer 1

The probability of different outcomes for flawed silk scarves out of a sample of 30 can be calculated using the binomial distribution formula. This involves finding the probability of exactly one flawed scarf, none, and more than three by applying the formula with n=30 and p=0.005.

Step-by-step explanation:

The subject of this question is Mathematics. Specifically, this involves the application of probability theory to real-world scenarios. To calculate the probabilities of different outcomes for silk scarves with flaws, we will use the binomial distribution formula given the small probability of a defect (0.5%) and the finite number of trials (30 scarves).

For a binomial distribution, the probability of getting exactly k successes in n trials can be calculated with the formula:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

Where:

C(n, k) is the combination of n items taken k at a time.

p is the probability of success on any given trial.

X is the random variable representing the number of successes (flawed scarves).

Calculating Probabilities

Exactly One Flawed Scarf: Calculate P(X = 1) using the formula with n = 30 and p = 0.005

No Flawed Scarves: Calculate P(X = 0)

More than Three Flawed Scarves: Since calculating for each possible number greater than three can be cumbersome, consider calculating the complement (P(X ≤ 3)) and subtracting from 1: 1 - P(X ≤ 3)

The last calculation involves adding the probabilities of 0, 1, 2, and 3 flawed scarves using the binomial formula and finding the complement.

Answer 2

The binomial probability is determined by the formula

`\begin{gathered} P_x=\binom{n}{x}p^xq^(n-x) \\ \text{where} \\ P\text{ is the binomial probability} \\ x\text{ is the number of times for a specific outcome within }n\text{ trials} \\ p\text{ is the probability of success on a single trial} \\ q\text{ is the probability of failure on a single trial} \\ n\text{ is the number of trials} \end{gathered}`

Here, we will assume that finding the flaw is the "success" of the trial hence the following given

`\begin{gathered} p=0.05 \\ q=0.95 \\ n=30 \end{gathered}`

Finding the probability of one flawed scarf

`\begin{gathered} P(X=1)=\binom{30}{1}(0.05)^1(0.95)^(30-1) \\ P(X=1)=0.3389 \end{gathered}`

Finding the probability of no flawed scarves

`\begin{gathered} P(X=0)=\binom{30}{0}(0.05)^0(0.95)^(30-0) \\ P(X=0)=0.2146 \end{gathered}`

Finding the probability of more then three flawed scarves

`\begin{gathered} P(X>3)=1-P(X\le3) \\ P(X>3)=1-0.93922843869 \\ P(X>3)=0.0608 \end{gathered}`