Biased Coins
When two fair coins are tossed, then getting 1 Head is twice as likely as either getting 2 Heads or no Heads, at all.
Can there be two biased coins, such that getting no heads, one head, or two head are equally likely?
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
For the two head and two tails to be equally likely, if one coin is biased toward head, the other needs to be biased toward tails to the same degree.
If the likelihood of the first coin getting heads is a , then likelihood of two heads is a ( 1 − a ) .
Likelihood of heads coming up once is then a 2 + ( 1 − a ) 2 .
So we get an equation a ( 1 − a ) = a 2 + ( 1 − a ) 2 ,
which simplified to 3 a 2 − 3 a + 1 = 0
This equation does not have a real solution, so the situation cannot occur.