smith is a weld inspector at a shipyard. He knows from keeping track of good and substandard welds that for a afternoon shift 5% of all welds done will be substandard. Of smith checks 300 of the 7500 welds completed that shift, what is the probability that he will find less than 20 substandard welds?
0.9066
87divid5
To find the probability that Smith will find less than 20 substandard welds out of the 300 welds he checks, we need to use the binomial distribution.
The formula for calculating the probability of x successes in n trials, with a probability of success p, is:
P(X = x) = (nCx) * p^x * (1-p)^(n-x)
Where:
- n is the number of trials (300 in this case)
- x is the number of desired successes (less than 20 substandard welds in this case)
- p is the probability of success (5% or 0.05 in this case)
- nCx is the number of combinations of n items taken x at a time
First, let's calculate the probability of one substandard weld:
P(X = 1) = (300C1) * 0.05^1 * (1-0.05)^(300-1)
Using the combination formula (nCx) = n! / (x! * (n-x)!) where "!" represents the factorial:
P(X = 1) = (300! / (1! * (300-1)!)) * 0.05 * 0.95^299
Now using a calculator or software that can handle large factorials, calculate this value:
P(X = 1) ≈ 0.2514
Repeat this calculation for all possible values of x less than 20 and sum up the probabilities:
P(X < 20) = sum of P(X = x) for x = 0 to 19
You can use a calculator or software to calculate this sum.
To find the probability that Smith will find less than 20 substandard welds out of the 300 welds inspected, we can use the binomial distribution formula.
The binomial distribution formula is:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes (substandard welds in this case)
- n is the total number of trials (welds inspected)
- k is the number of successes (substandard welds inspected)
- p is the probability of success (probability of a substandard weld)
In this case, the total number of trials (welds inspected) is 300, and we want to find the probability of finding less than 20 substandard welds, which means we need to calculate the probabilities for k = 0, 1, 2, ..., 19.
Since we know that 5% of all welds done will be substandard, the probability of getting a substandard weld (p) is 0.05.
Now, let's calculate the probabilities for each possible number of substandard welds inspected (k) and sum them up to find the final probability.
P(X < 20) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 19)
P(X = k) = C(300, k) * 0.05^k * (1-0.05)^(300-k)
To calculate the probability for each value of k and sum them up, you can use a calculator, spreadsheet software, or statistical software like Excel or Python.
Alternatively, you can use online binomial distribution calculators or statistical packages to directly find the probability without doing individual calculations.