# If the probability that an alpha will deflect is $1/10000$, for $n$ layers, is the probability is only $1/10000n$?

I have attached a picture of an extract I read on Wikipedia (also in the AQA A-Level Physics specification and textbook). It says that 1/10000 alpha particles deflected in the alpha particle scattering experiment and therefore, the probability is 1/10000 that it will deflect (more than 90 degrees). It then goes on to say that for n layers of atoms, the probability is 1/10000n. This doesn't make sense to me. If the number of layers increases, that means the probability of deflection is getting smaller.

Surely if there are more layers, deflection is more likely?

• I've deleted some comments which were answering the question. Please keep in mind that comments are meant for suggesting improvements or requesting clarification on the question, not for answering. – David Z Feb 11 at 20:30

Surely if there are more layers, deflection is more likely?

If you know the probability of a single layer deflecting and you add more layers, then yes the probability of the foil deflecting goes up.

We could even write something like $$P_{foil} \approx P_{layer} \times \text{n}$$

But in the experiment, we don't know the single layer probability and we we do know the total deflection from the foil, so we change the equation: $$P_{layer} \approx \frac{P_{foil}}{n}$$

It says that the probability of a single layer deflection goes down as the number of layers increases given a constant level of detection from the foil

• @Alice if one of the answers has helped you, please be sure to accept it by clicking the green checkmark next to it. – BowlOfRed Feb 14 at 6:50

If the probability of deflection by one layer is $$p$$, the probability of deflection by $$n$$ layers is $$1-(1-p)^n$$. As $$n$$ goes to infinity, this goes to 1. For small $$p$$ and large $$n$$, a good approximation is $$1-e^{-pn}$$.

However, for small $$p$$ and small $$n$$, a good approximation is $$pn$$, which is the approximation Wikipedia is using. Note that it is saying that the probability of deflection by one layer is $$1/10000n$$ and the probability of deflection by $$n$$ layers is $$1/10000$$, not the other way around as you thought.