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The ferry crosses the strait from beach A to beach B at regular intervals of 20 minutes. The ferry has room for 7 cars.

Suppose no cars left behind when ferry first leave. Suppose the number of cars arriving on the beach A the next 20 minutes in the following cases is the Poisson distribution parameter 6.

What is the probability that the ferry will be full the next time it leaves at beach A?

Should i use poisson distribution and is lambda on this case the 6?

Poisson distribute

Matthew Towers
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pokeri
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  • Yes you can use the Poisson distribution with $\lambda=6$, though I would say this could tell you how likely the ferry is to be full when it next leaves beach $A$ rather than when it arrives – Henry Sep 17 '19 at 18:14
  • Yes you are right i made mistake it should be when it next leaves the beach. Should i use the normal poisson distribution? – pokeri Sep 17 '19 at 18:16
  • That would work, so long as you look at the correct probability – Henry Sep 17 '19 at 19:03
  • Is there any examples what formula i need to use i can't get the right answer with anything? – pokeri Sep 18 '19 at 13:40
  • What do you think is the correct answer and how did you calculate it? – Henry Sep 18 '19 at 16:36
  • I tried to calculate it with poisson distribute formula and i get 0.104 but that's not right. I added the formula to the original post. – pokeri Sep 18 '19 at 17:02
  • You seem to have calculated the probability for $k=8$. I would have thought you need to consider $k \ge 7$. This might suggest an infinite sum, so an alternative calculation is $1$ minus the probability the number is $k=0$, $1$, $2$, $3$, $4$, $5$ or $6$. – Henry Sep 18 '19 at 17:27

1 Answers1

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\begin{align} \mathbb P(X\geqslant 7) &= 1-\mathbb P(X<6)\\ &= 1-\sum_{k=0}^6 \mathbb P(X=k)\\ &= 1-\sum _{k=0}^6 \frac{6^k}{k!}e^{-6}\\ &=1-\frac{1223}{5}e^{-6}\approx 0.3936972. \end{align}

Math1000
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