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Probability is just the way in which likely something is to occur. At the point when we're uncertain about the result of an occasion, we can discuss the probabilities of specific results — how likely they are. The investigation of occasions represented by likelihood is called insights.

We know,

Time is x= $10$minutes

Decay rate m = $\frac{12}{60}=0.2$

Using,

$P(x>10)=1−P(x<10)$

The probability that the duration between two successive visits to the website is more than ten minutes is,

$$\begin{gathered} P(x>10)=1−\left(1−e^{−0.2×10}\right) \\ P(x>10)=0.1353 \end{gathered}$$

We know,

Decay rate $m=\frac{12}{60}=0.2$

The top $25\%$of durations between visits are calculated,

$$\begin{gathered} P(x<k)=1−e^{−0.2×k} \\ 0.25=1−e^{−0.2×k} \end{gathered}$$

Using log,

$\mathrm{In}\left({\mathrm{e}}^{−0.2×\mathrm{k}}\right)=\mathrm{In}\left(0.75\right)$

$\mathrm{k}=1.4385$

Hence the duration is$1.4385min$

We know,

The probability that the next visit will occur within the next $5$minutes ,

$P(20<x<25)=P(x<25)−P(x<20)$

$$\begin{gathered} \mathrm{P}(20<x<25)=1−e^{−0.2×25}−\left(1−e^{−0.2×20}\right) \\ \mathrm{P}(20<x<25)=0.0183−0.0067 \\ =0.0116 \end{gathered}$$

We know,

The probability that fewer than $7$visits occur within a one-hour period is calculated using the Poisson distribution,

$⇒\mathrm{P}(x<7)=P(x≤6)$

Using the calculator we find $P(x≤6)$,

$Poissoncdf(12,6)=0.458$