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In a aloha implemented shared channel probability of transmission of a station in a time span of T is p. Given, probability such that NO station transmit in a time duration of $2T$ is $50\%$ , where T = one frame transmission time. What is the value of p is if total no of station = $100$
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I did in the following way,

For any time span probability of there being x transmission attempts is

$$\frac{A^xe^{-A}}{x!}$$

Where A is the average no of transmission attempt in that time span.

In that question time duration is $2T$ . And we know that channel load G is the average transmission attempt in T time duration. So, in 2T time the average transmission attempts will be 2G.

SO, plugging A = 2G in the above formula,

$$\frac{(2G)^xe^{-2G}}{x!}$$

For the given condition , No transmission attempts is 2T  => x = 0

Probability = $\frac{(2G)^0e^{-2G}}{0!}$ = 0.5

=> G = 0.3465

Now for poisson Mean of average is  = Np

=> p = 0.003465 (very low)

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