Rhine paradox - discrepancies

Question

In the Rhine paradox described here, the author arrives at a probability of 0.98 that at least 1 in 1,000 people would guess at ten cards right.

My calculations seem to produce a different result. This is how I did it:

$\text{P(at least one person getting all 10 cards right)} \\ = 1 - \text{P(no person getting all 10 cards right)} \\ = 1 - \text{P(person 1 not getting all cards right} \\ \text{* person 2 not getting all cards right *}\ldots \\ \text{* person 1000 not getting all cards right)} \\ = 1 - (((2^{10})-1/(2^{10})) * ((2^{10})-1/(2^{10})) * \ldots * ((2^{10})-1/(2^{10}))) \\ = 1 - ((0.9990234375)^{1000}) \\ = 1 - 0.37642379805 \\ = 0.62$

So this gives only a probability of 0.62, as opposed to 0.98, that at least 1 person out of 1000 will get all ten cards right.

What did I do wrong?

Answer 1

Assuming the 10 cards are picked randomly, your answer seems correct but a little cumbersome. The probability that one person wins is $$\frac{1}{2}^{10}$$

The probability that at least one person wins:

$$\begin{align*} P(\text{at least one wins}) &=1-P(\text{no one wins})\\ &=1-\left(1-\frac{1}{2}^{10}\right)^{1000}\\ &\approx 0.6235762 \end{align*}$$

This can be verified in R:

sum(dbinom(1:1000, 1000, .5^10))

0.6235762