Smith is a weld inspector at a shipyard. He knows from keeping track of good and substandard welds that for
the afternoon shift 5% of all welds done will be substandard. If Smith checks 300 of the 7500 welds completed
that shift, what is the probability that he will find between 10 and 20 substandard welds?
0.4066
0.8132
To find the probability that Smith will find between 10 and 20 substandard welds, we can use the binomial probability formula.
The formula for binomial probability is:
P(X = k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes
- n is the total number of trials
- k is the number of successful outcomes
- p is the probability of a successful outcome in each trial
- (n C k) represents the number of ways to choose k successes from n trials
In this case, n = 300 (the number of welds Smith checks), k can be any value between 10 and 20 (the number of substandard welds), and p = 0.05 (the probability of a weld being substandard).
Now, let's calculate the probability:
P(10 <= X <= 20) = P(X = 10) + P(X = 11) + ... + P(X = 20)
P(X = k) = (300 C k) * (0.05)^k * (1-0.05)^(300-k)
Let's calculate each individual term and sum them up:
P(X = 10) = (300 C 10) * (0.05)^10 * (1-0.05)^(300-10)
We can substitute the values into the formula to find the probability of Smith finding exactly 10 substandard welds:
P(X = 10) = (300 C 10) * (0.05)^10 * (0.95)^290
Using a calculator, we can compute this value. Repeat the calculation for each value of k between 10 and 20, and sum up the probabilities to find the final result.
To find the probability that Smith will find between 10 and 20 substandard welds, we can use the binomial probability formula. The binomial distribution is used when there are only two possible outcomes (in this case, good or substandard welds), and the probability of success and failure remains constant.
Let's break down the information given:
- The total number of welds completed during the afternoon shift is 7500.
- Smith checks 300 welds.
Now, we can calculate the probability using the following steps:
1. Determine the probability of finding a substandard weld:
- The given information states that 5% of all welds done will be substandard.
- Since the probability of finding a substandard weld is 5%, the probability of finding a good weld is 1 - 0.05 = 0.95.
2. Calculate the expected number of substandard welds Smith would find:
- The proportion of substandard welds is given as 5% or 0.05.
- The expected number of substandard welds is calculated as the proportion multiplied by the total number of welds checked: 0.05 * 300 = 15.
3. Calculate the probability of finding between 10 and 20 substandard welds:
- Using a binomial probability calculator or a statistical software, we can calculate the probability of finding between 10 and 20 substandard welds using the following formula:
P(X=k) = nCk * p^k * (1-p)^(n-k)
where n is the total number of trials (300), k is the number of successful trials (substandard welds), p is the probability of success (0.05), and (1-p) is the probability of failure (0.95).
- Calculate the probabilities for k=10, 11, 12, ..., 20.
- Add up the individual probabilities to get the final probability.
Alternatively, you can use statistical software, online calculators, or programming languages with statistical packages (such as Python with NumPy and SciPy libraries) to find the probability directly without manual calculation.
Please note that the calculation involves combinatorics, so using a calculator or software might be more efficient and accurate.