Why my calculation is different from R calculation A radioactive object emits pa

Question

Why my calculation is different from R calculation

A radioactive object emits particles according to a Poisson process at an
average rate of 7.5 particles per second. We observe the object for a total of 5 seconds.
Suppose it is known that at least 20 particles will be emitted during this interval, what is
the probability that no more than 25 particles will be emitted during this interval?

P( X = 20 ) = e ^-37.5 * 37.5^20 / 20! = 0.0006
P( X = 21 ) = e ^-37.5 * 37.5^21 / 21! = 0.0011
P( X = 22 ) = e ^-37.5 * 37.5^22 / 22! = 0.002
P( X = 23 ) = e ^-37.5 * 37.5^23 / 23! = 0.0032
P( X = 24 ) = e ^-37.5 * 37.5^24 / 24! = 0.005
P( X = 25 ) = e ^-37.5 * 37.5^25 / 25! = 0.0075

Explanation / Answer

The calculations are correct , it is just the rounding of numbers that effects the final answers .

however , if you look closely rounding the answers to 3 decimal places for both the values , the answer is

0.019

0.0194 ~ 0.019

0.018772 ~ 0.019

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