The average number of radioactive particles passing through a counter during 1 millisecond in a lab experiment is 4. What is the probability that more than 2 milliseconds pass between particles?
What is the variability of this count? More information needed.
I've been working this out and can't get the answer shown in the book...I use mu = 1/4 thus lambda = 4 P(X > 2) = 1-(1-F(x)) = e^-4x = e^-8 This is not the answe given however so I am lost on this one.
To find the probability that more than 2 milliseconds pass between particles, we can use the exponential probability distribution.
The probability density function (pdf) of the exponential distribution is given by:
f(x) = λ * e^(-λ*x)
Where λ (lambda) is the rate parameter of the distribution.
In this case, the average number of particles passing through the counter during 1 millisecond is 4. Since the distribution is exponential, we can calculate the rate parameter λ as 1/mean (λ = 1/4).
Now, let's calculate the probability that more than 2 milliseconds pass between particles.
P(X > 2) = ∫[2, ∞] λ * e^(-λ*x) dx
Using the cumulative distribution function (CDF) of the exponential distribution, the probability can be calculated as:
P(X > 2) = 1 - P(X ≤ 2)
P(X > 2) = 1 - ∫[0, 2] λ * e^(-λ*x) dx
P(X > 2) = 1 - (1 - e^(-2*λ))
Substituting the value of lambda, we get:
P(X > 2) = 1 - (1 - e^(-2*(1/4)))
P(X > 2) = 1 - (1 - e^(-1/2))
P(X > 2) = 1 - (1 - 0.6065)
P(X > 2) = 0.6065
Therefore, the probability that more than 2 milliseconds pass between particles is approximately 0.6065, or 60.65%.
To calculate the probability that more than 2 milliseconds pass between particles, we need to first determine the average time between each particle.
The average number of radioactive particles passing through the counter during 1 millisecond is 4. This implies that on average, the time between each particle is 1/4 milliseconds.
To find the probability that more than 2 milliseconds pass between particles, we need to calculate the probability of having a time interval greater than 2 milliseconds.
The probability of this can be determined using exponential distribution, since the times between particles in this experiment are modeled as an exponential random variable.
The exponential distribution probability density function (pdf) is given by the formula:
f(x) = λ * exp(-λ * x),
where λ is the rate parameter (in this case, the reciprocal of the average time between particles) and x is the time between particles.
In this scenario, the rate parameter λ = 1/(1/4) = 4. Therefore, the pdf becomes:
f(x) = 4 * exp(-4 * x).
To find the probability that more than 2 milliseconds pass between particles, we need to calculate the integral of the exponential distribution pdf from 2 to infinity:
P(X > 2) = ∫[2, ∞] f(x) dx = ∫[2, ∞] 4 * exp(-4 * x) dx,
where P(X > 2) represents the probability that more than 2 milliseconds pass between particles.
Calculating this integral will provide the desired probability.