# probability of 10 heads from 1000 coin tosses

The probability of 10 heads if you toss a fair coin 10 times is

$$P(10H) = (1/2)^{10} = 0.1 \%$$

What is the probability of some coin getting 10 heads if you toss 1000 fair coins 10 times each ?

• At least 10 heads out of 10 000 tosses? At most 10? Exactly 10? Commented Nov 23, 2015 at 8:01
• Out of the 1000, at least 1 coin getting exactly 10 heads. Commented Nov 23, 2015 at 9:21
• Is this a homework? If so, please add [self-study] tag and refer to stats.stackexchange.com/tags/self-study/info
– Tim
Commented Nov 23, 2015 at 9:35
• Not homework, this was discussed in class. But the self-study tag does seem appropriate. Commented Nov 23, 2015 at 9:42
• OK, thanks! Btw, notice that this page supports $\TeX$ formatting (e.g. en.wikibooks.org/wiki/LaTeX/Mathematics or ftp.ams.org/pub/tex/doc/amsmath/short-math-guide.pdf) - I edited your post.
– Tim
Commented Nov 23, 2015 at 10:04

$P(\text{a coin get 10 heads}) = 0.5^{10}$

$P(\text{a coin doesn't get 10 heads}) = 1 - 0.5^{10}$

$P(\text{all coins don't get 10 heads}) = (1 - 0.5^{10})^{1000}$

$P(\text{at least one coin gets 10 heads}) = 1 - (1 - 0.5^{10})^{1000} \approx 0.624$

• for formatting formulas you can use $\TeX$ (I edited your answer). By the way, could you please describe why you consider this approach correct?
– Tim
Commented Nov 23, 2015 at 9:34
• P(no coins get all heads) + P(some coins get all heads) = 1. And you can reformulate that as P(no coin get all heads) + P(at least 1 coin get all heads) = 1 Commented Nov 23, 2015 at 9:43
• That is why you should add more details to your answer because it is not clear.
– Tim
Commented Nov 23, 2015 at 9:45
• Well, often it is enough just to give a hint and then everything become clear. Or is it against site policy to give hints? Commented Nov 23, 2015 at 9:56
• It is not against, in fact it is totally how self-study questions should be answered (see stats.stackexchange.com/tags/self-study/info). However, for your answer to be usable for other users and other questions you should provide a description that enables reader to extrapolate your answer to other problems.
– Tim
Commented Nov 23, 2015 at 9:59