Poisson distribution with big numbers

Question

I failed this question at an exam and would like help in explaining how to solve it: Historical data show that approximately 70 000 vehicles cross Älvsborgsbron every day. What is the probability that more than 7 500 000 vehicles cross Älvsborgsbron in 100 days? For full points you must motivate and explain all assumptions and approximations.

Thanks for the help

Edit: I have tried to use the poisson formula and end up with values that my calculator says doesnt work. I¨ve also tried this \frac{\left(7500000-7000000\right)}{2645.75} but gives my a result of 188,98. So doesnt make sense as I should get a percentage answer. My only though about that number is that it is so high that its basically zero percent chance of happening.

Answer 1

You're on the right track.

1. Assume a normal distribution (see here why)
2. Assume a daily standard deviation - let's say $\sigma$ daily is 10,000.
3. Calculate the Z-score for 7,500,000 crossings $$ Z = (7,500,000 - 7,000,000)/100,000 $$

I think your assumption in step 2, where you've assumed a variance of 7,000,000 is wrong and this has led to a very high Z-score. Your conclusion seems right - there's an extremely low chance of this occurring.