Question

of the people passing through an airport metal detector, 0.1% activate it, Let X= number among a random selected group of 500 who activate the detector. what is the probability that exactly two people would activate the detector.

Answer 1

i need help in solving this question

Answer 2

Hint: you have two possible outcomes (activation or non-activation of the detector), so this must involve some sort of binomial probability distribution.

You're specifically given a "success" probability (i.e. probability of activation) of \(0.1\%=0.001=\dfrac{1}{1000}\). You're also given a number of trials (500 people pass through the detector) and you want to find the probability of *exactly* 2 people activating the detector.

Recall the binomial probability formula:
\[P(X=k)=\binom nk p^{1-k}(1-p)^k\]
where in this case, \(k=2\), \(n=500\), and \(p=\dfrac{1}{1000}\).

Answer 3

It may be more practical to use the normal distribution in this case. Since the sample is large (500 is fairly large). The mean and variance of a binomial distribution are given by \(\mu=np\) and \(\sigma^2=np(1-p)\), so in this case you have \(\mu=\dfrac{1}{2}\) and \(\sigma^2=\dfrac{999}{2000}\).

This means the binomial distribution, defined for the situation as
\[f_X(x)=\begin{cases}\dbinom nx p^(n-x)(1-p)^x&\text{for }0\le x\le n\\\\
0&\text{otherwise}\end{cases}\]
can be approximated by the normal distribution with the same mean and variance:
\[f_X(x)\approx \frac{1}{\sqrt{2\pi\times\dfrac{999}{2000}}}\exp\left(-\frac{\left(x-\dfrac{1}{2}\right)^2}{2\times\dfrac{999}{2000}}\right)\]
The nice thing about continuous distributions is that probabilities of picking exact alues are essentially zero. You can verify this by computing the binomial probability with a calculator.

Answer 4

thank you so much
i have one more question to ask, may I??

Answer 5

Fell free, but I have to get going soon. I can come back later. If you're pressed for time, try posting another question and someone else may try to help you out.

Answer 6

An examination consists of a large number of question of the multiple choice type, with each question having five possible answers but only one of the five being the correct answer. If a student receives 3 points for each correct answer and-1 point for each incorrect answer and if on each of ten question, his probability of guessing the correct answer is 1/3, what Sis the probability of obtaining a total positive score on those ten questions

Answer 7

An examination consists of a large number of question of the multiple choice type, with each question having five possible answers but only one of the five being the correct answer. If a student receives 3 points for each correct answer and-1 point for each incorrect answer and if on each of ten question, his probability of guessing the correct answer is 1/3, what Sis the probability of obtaining a total positive score on those ten questions

Answer 8

the total number of hours measured in units of 100 hours that a family runs a vaccum cleaner over a period of one year is a continuous random variable, x, that has probability density function:
f(X) =
x, 0≤ x ≤1
k-x, 1>x>2
0 elsewhere
(a) determine that value of k
(b) find the probability that over a period of one year, a family runs their vaccum cleaner less than 110 hours.
also find probability: P [ 0.5≤x ≤ 1.5

Answer 9

the total number of hours measured in units of 100 hours that a family runs a vaccum cleaner over a period of one year is a continuous random variable, x, that has probability density function:
f(X) =
x, 0≤ x ≤1
k-x, 1>x>2
0 elsewhere
(a) determine that value of k
(b) find the probability that over a period of one year, a family runs their vaccum cleaner less than 110 hours.
also find probability: P [ 0.5≤x ≤ 1.5