Question

The concentration of particles in a suspension is 30 per mL. One hundred 2 mL samples are drawn. What is the probability that at least 90 of them contain more than 50 particles?

Answer 1

The problem at hand is a statistical probability question from college-level mathematics. It requires additional information on the statistical distribution or assumptions for the central limit theorem to accurately calculate the probability of 90 out of 100 samples containing more than 50 particles. Without this information, an exact solution cannot be confidently provided.

Step-by-step explanation:

The student's question involves calculating the probability that a certain number of samples from a suspension will contain a specific concentration of particles. This is a problem that deals with statistical distributions and sampling theory, which falls under the category of mathematics, particularly in the college-level subject of statistics or probability theory.

Without the exact statistical distribution of the particle counts in the samples, such as a normal distribution or Poisson distribution, it's challenging to calculate the exact probability requested. The information provided in the references does not directly relate to solving the problem at hand, which requires more details about the distribution of the particle counts or an application of the central limit theorem if the assumptions for its use are met.

In statistical problems like this, it's common to use the central limit theorem to approximate the distribution of sample means when the sample size is sufficiently large. However, without further information regarding the underlying distribution of the particle counts or a clear statement that the central limit theorem applies, we cannot confidently provide an accurate answer to this probability question.

Answer 2

To calculate the probability of at least 90 of the 100 2 mL samples containing more than 50 particles, we can use the binomial distribution formula. The probability of success for each sample is 0.03 (30/1000), and we can calculate the probability using the formula P(X >= 90) = 1 - P(X < 90).

Step-by-step explanation:

To calculate the probability that at least 90 of the 100 2 mL samples contain more than 50 particles, we can use the binomial distribution formula. Let X be the random variable representing the number of samples that contain more than 50 particles. The probability of each sample containing more than 50 particles is given by the concentration of particles in the suspension, which is 30 per mL. Therefore, the probability of success (p) is 30/1000 = 0.03. The number of samples drawn (n) is 100, and we want to find the probability of at least 90 of them containing more than 50 particles. Using the binomial distribution formula, we can calculate the probability as follows:

P(X >= 90) = 1 - P(X < 90) = 1 - P(X = 0) - P(X = 1) - ... - P(X = 89)

Calculating this sum using the binomial distribution formula will give us the probability that at least 90 of the samples contain more than 50 particles.