Let lim _{t rightarrow 0}(1+5 t)^{frac{1}{t}} | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given that $\lim _{t \rightarrow 0}(1+5t)^{\frac{1}{t}} = K$. Using the standard limit $\lim _{t \rightarrow 0}(1+at)^{\frac{1}{t}} = e^a$, we get

Vedclass · Accepted Answer

$\frac{K-1}{K}$

Vedclass · Answer

(B) Given that $\lim _{t ightarrow 0}(1+5t)^{\frac{1}{t}} = K$. Using the standard limit $\lim _{t ightarrow 0}(1+at)^{\frac{1}{t}} = e^a$, we get $K = e^5$. For the random variable $X$ representing the number of successes in $n = 100$ independent trials with probability of success $p = 0.05$, the distribution follows a Binomial distribution $B(n, p)$. Since $n$ is large and $p$ is small, we can approximate this using a Poisson distribution with parameter $\lambda = np = 100 	imes 0.05 = 5$. The probability of getting at least one success is $P(X \geq 1) = 1 - P(X = 0)$. Using the Poisson formula $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$, we have $P(X = 0) = e^{-5}$. Therefore, $P(X \geq 1) = 1 - e^{-5} = 1 - \frac{1}{e^5} = 1 - \frac{1}{K} = \frac{K-1}{K}$.

Let $\lim _{t \rightarrow 0}(1+5 t)^{\frac{1}{t}}=K$ and $X$ be the random variable representing the number of successes in $100$ independent trials. If the probability of success in each trial is $0.05$, then the probability of getting at least one success is

Similar Questions

An experiment succeeds twice as often as it fails. The probability of at least $5$ successes in the $6$ trials of this experiment is

What is the probability of getting an even number exactly $3$ times when a die is thrown $5$ times?

If the sum of mean and variance of a Binomial Distribution is $\frac{15}{2}$ for $10$ trials,then the variance is (in $.5$)

$A$ rifle man is firing at a distant target and has only $10\%$ chance of hitting it. The minimum number of rounds he must fire in order to have $50\%$ chance of hitting it at least once is

$A$ medicine for fever can cure $75\%$ of the people. Three people suffering from fever use this medicine. What is the probability that all three people are cured?

Vedclass Test Series

Exam Paper Generator

Online Exam Module