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The number of planet transit discovered for every 3000 stars follows a Poisson distribution with$位=5$

The probability that X represents the number of successes of a random variable in Poisson process is given by:

$P\left(X=x\right)=\frac{e^{-位}{位}^x}{x!}$

Where,

X is the number of successes

$e=2.71828$is the Euler number

$位$is the mean in the given time interval.

We have $位=5$

We have to calculate the probability that, in the next 3,000 stars monitored by the Kepler mission, more than 10 planet transits will be seen that is $P\left(X>10\right)$.

We can see either 10 or less than 10 planets, or we can see more than 10 planets.

We know that the sum of all the probabilities is 1.

Hence,

$$\begin{gathered} P\left(X鈮�10\right)+P\left(X>10\right)=1 \\ P\left(X>10\right)=1-P\left(X鈮�10\right) \end{gathered}$$

Where,

$$\begin{gathered} P\left(X鈮�10\right)=P\left(x=0\right)+P\left(x=1\right)+P\left(x=2\right)+P\left(x=3\right)+P\left(x=4\right)+ \\ P\left(x=5\right)+P\left(x=6\right)+P\left(x=7\right)+P\left(x=8\right)+P\left(x=9\right)+P\left(x=10\right) \end{gathered}$$

Hence, from the formula of Poisson distribution:

$$\begin{gathered} P\left(X=x\right)=\frac{e^{-位}{位}^x}{x!} \\ P\left(X=0\right)=\frac{e^{-5}{位}^0}{0!} \\ =0.0067 \\ P\left(X=1\right)=\frac{e^{-5}{位}^1}{1!} \\ =0.0337 \\ P\left(X=2\right)=\frac{e^{-5}{位}^2}{2!} \\ =0.0842 \\ P\left(X=3\right)=\frac{e^{-5}{位}^3}{3!} \\ =0.1404 \\ P\left(X=4\right)=\frac{e^{-5}{位}^4}{4!} \\ =0.1755 \\ P\left(X=5\right)=\frac{e^{-5}{位}^5}{5!} \\ =0.7154 \\ P\left(X=6\right)=\frac{e^{-5}位6}{6!} \\ =0.1462 \\ P\left(X=7\right)=\frac{e^{-5}{位}^7}{7!} \\ =0.1044 \\ P\left(X=8\right)=\frac{e^{-5}{位}^0}{0!} \\ =0.0067 \\ P\left(X=9\right)=\frac{e^{-5}{位}^9}{9!} \\ =0.0362 \\ P\left(X=10\right)=\frac{e^{-5}{位}^{10}}{10!} \\ =0.0181 \end{gathered}$$

Adding all the probabilities,

$$\begin{gathered} P\left(x鈮�10\right)=0.0067+0.0337+0.0842+0.1404+0.1755+0.1754+ \\ 0.1460+0.1043+0.06520+0,0362+0.0181 \\ P\left(x鈮�10\right)=1-P\left(x鈮�10\right) \\ =1-0.9658 \\ =0.0342 \end{gathered}$$

Hence, there is 3.4% probability that, in the next 3000 stars monitored by the Kepler mission, more than 4 planet transits will be seen .