Calculate Statistic False Positives with Our Examples

Wire manufactured by a company is tested for strength. The test gives a correct positive result with a probability of 0.85 when the wire is strong, but gives an incorrect positive result (false positive) with a probability of 0.04 when in fact the wire is not strong.
If 98% of the wires are strong, and a wire chosen at random fails the test, what is the probability it really is not strong enough?

A disease effects 1/1000 newborns and shortly after birth a baby is screened for this disease using a cheap test that has a 2% false positive rate (the test has no false negatives). If the baby tests positive, what is the chance it has the disease?
I've got $P(\text{disease | positive})=\frac{P(d^p)}{P(p)}=\frac{(1/1000)}{(1/1000+2/100\ast <!--\; \ast \; -->999/1000)}=\frac{1}{(20+49/50)}$
Is this right?

Given that out of $1000$ individuals where $60$ use a drug. I am given the probability of a false positive is $0.009$ and the probability of a false negative is $0.10$. I'm trying to find the true positive and true negative.
Let $D$ be the even that the user uses a drug and $D^C$ the event the user is not a drug user. I know that $P(D)=0.06$ and $P(D^C)=0.994$ as well as $P(+|D^c)=0.009$ and $P(-<!--\; -\; -->|D)=0.10$. So that means that $P(+|D^C)=0.009⟹<!--\; ⟹\; -->0.009\ast <!--\; \ast \; -->990=8.91$ people. Also $P(-<!--\; -\; -->|D)=0.10⟹<!--\; ⟹\; -->60\ast <!--\; \ast \; -->.10=6$ people. Does it follow that $P(-<!--\; -\; -->|D^C)=981.09/990=0.991$ and $P(+|D)=54/60=0.90$?

A town with a population 10,000 has suffered an outbreak of dragon pox, with 3% of the population being infected. There is a test to diagnose the disease. If you have the dragon pox, the test will correctly register positive 97% of the time and will yield a false negative 3% of the time. If you do not have the dragon pox the test will register that correctly 97% of the time and will yield a false positive 3% of the time. Given that you tested positive for the disease, what is the probability that you actually have dragon pox?

5% of the population have the disease (D). A test is available that has a 10% false positive and a 10% false negative rate.
part (B) of the question asks what is the probability of having the disease given that you test positive. I'm quite confident the answer to this is ~ 32%
Part (D) - Suppose that there is no cost or benefit from testing negative but a benefit B from true positives which detect the disease and a cost C to false positives.
i) What is the expected value of the testing programme?
ii) What benefit-cost ratio would you require to proceed with the testing programme?
I'm not quite sure to how to answer either of part D. My somewhat educated guesses are
the expected value = 0.045B - 0.095C
and the benefit to cost ratio required to proceed with the testing programme is 2.11 (or greater)

if 1:1000 of people is sick. the probability to be false positive is 0.07. if a person is sick there is not chance the test for the disease is wrong. If someone random is got a positive result, what are the chances he's actually sick?
I got 1.5% and wanted to check because I feel it should be more since the diagnose is never wrong. I took 1/1000 and divided it by 0.07+1/1000 and multiplied by 100 to get the percent.

For what composite numbers $x$ will $a^{x-<!--\; -\; -->1}\equiv <!--\; \equiv \; -->1(\mathrm{mod}x)$ for $a\in <!--\; \in \; -->[2,n]$?
Can we generate $x$s that give false positives to the Rabin-Miller test for the first, say $10$, consecutive integers $a>1$?

While it makes some sense, it's not clear to me why those are different. If a test, say medical test, is correct 90% of time then chances of it being wrong is 10%.
But I've read things that say in medical field test with high accuracy for negatives are used as screening method and more expensive tests which are high accuracy positive tests are used only when former tests come positive. Because if former came negative we are confident patient doesn't have a disease but if it comes positive we are not as sure, thus we do expensive test. Which of course, makes sense.
I get there are 4 events:
1. Test is +, patient has a disease
2. Test is -, patient doesn't have a disease
3. Test is +, patient doesn't have a disease
4. Test is -, patient has a disease

A hospital identifies patients with an average false positive rate of 0.01. If it checks 31333 patients and returns a total of 357 positive results, then what's the expected number of true positives?

According to my research the Fowlkes-Mallows Index should be a good tool to get a comparable evaluation score, but I am not sure about the correct application.
Currently I calculate the true positives, false positives and false negatives for each cluster, then sum them together for total values and add them in the formula.
$FM=\sqrt{\frac{TP}{TP+FP}\cdot <!--\; \cdot \; -->\frac{TP}{TP+FN}}$
where $TP$ is the number of true positives, $FP$ is the number of false positives, and $FN$ is the number of false negatives.
Is this the correct application of the Algorithm?

Consider the generator polynomial $100101$ and the data $1011100110001$. Applying the CRC algorithm we get the transmitted message:
$\underset{Data}{\underset{⏟<!--\; ⏟\; -->}{1011100110001}}\underset{Remainder}{\underset{⏟<!--\; ⏟\; -->}{00011}}$
I need to make 3 errors in the transmitted word (that is, bit flips) that the receiver won't notice. Hence, the CRC algorithm will fail to notice the errors and will falsely the data. How can this be arranged?

in order to remember stuff i need to understand their reason. Right now i cannot remember what is type 1 error and what is type 2 error why is the reason type 1 is false positive?

"A certain disease has an incidence rate of 2%. If the false negative rate is 10% and the false positive rate is 1%, compute the probability that a person who tests positive actually has the disease."
Why do we need to use Bayes' Theorem for this question?

Descartes rule of sign can be used to isolate the intervals containing the real roots of a real polynomial. The rule bounds the number of roots from above, that is, it is exact only for intervals having zero or one root. In methods like VCA, VAS and similar, it is used to count the number of sign changes to determine the number of roots.
My question is, what to do if the rule reports, say, two sign changes for interval which does not contain any roots?

True or false: if $a_n$ is any decreasing sequence of positive real numbers and $b_n$ is any sequence of real numbers converges to $0$, then $\frac{a_n}{b_n}$ diverges.

My problem is the following. I like to know if there exist a sentence true in complex a field but false in a field of positive characteristic.

How the false positive value affects accuracy?
$TP(truepositive)=2739$
$TN(truenegative)=103217$
$FP(falsepositive)=43423$
$FN(falsenegative)=5022$
$accuracy=\frac{TP+TN}{TP+TN+FP+FN}$
In this case the accuracy is $0.68$. Can I say that I have low accuracy because the value false positive is high? There is any relathion between false positive and the parameters true positive or true negative?

The latest worldwide virus has an infection rate of $0.1\mathrm{\%<!--\; \%\; -->}$ (that is, $1$ in $1000$ people actually have the virus). The chance that someone who has the virus tests positive is said to be $99\mathrm{\%<!--\; \%\; -->}$. The chance that someone who does not have the virus tests negative is also said to be $99\mathrm{\%<!--\; \%\; -->}$. What are the chances that someone who tests positive does not in fact have the virus (a “false positive”)?

3% of the population has disease X.
A laboratory blood test has
(a) 96% effective at detecting disease X, given that the person actually has it.
(b) 1% “false positive” rate. i.e, a person who does not have disease X has a probability of 0.01 of obtaining a test result implying they have the disease.
What is the probability a person has the disease given that the test result is positive?

A pregnancy test kit is 98.5% accurate for true positive result, i.e. the result is positive when the tester is actually pregnant. If she is not pregnant, however, it may yield a 0.8% false positive. Suppose a woman using this pregnancy kit is 60% at risk of being pregnant.
Not sure about her first test which turned out to be negative, the woman decides to take the test again. This second test, however, turns out to be positive. Assuming the two test are independent, find the probability that she is actually pregnant.
Now she is so confused whether or not she is pregnant. So she take the tests n more times and the results for these n more tests are all positive. Find the minimum value for n so that she can be at least 99.99% sure of pregnancy, assuming all test are independent.