Add new comment

Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.
Generating electricity without the use of fossil fuels is not just an engineering and industrial challenge, it is also a huge mathematical challenge.
In this podcast author Coralie Colmez shares insights into her novel The irrational diary of Clara Valentine.
We talk to early career mathematicians who spent some of their summer holiday solving problems posed by industry — such as how to blend a perfect smoothie!
Don't like plantbased meat alternatives, but want to spare animals and the environment? There's hope on the horizon, aided by a good helping of maths.
Inverse problems are mathematical detective problems. They can help solve crimes, are used in medical imaging, and much more.
write a and 2a for the amount of money in those two envelope, write x for the amount of money that is in your chosen envelope. The x can be a or 2a in both cases with the probability of 1/2. When the x is a, the amount of another envelope must be 2a, vice versa. If you change your mind to choose another envelope, in fact the expected amount you will get is:
"the expected value when your chosen envelope is a " plus "the expected value when your chosen envelope is 2a", which is 1/2*(2x) (when x equals a ) plus 1/2*x/2 (when x equals 2a).
so the expected amount you will get is 1/2*(2x(x=a)+x/2(x=2a))=1/2(2a+a)=3/2*a.
At the same time, the expected value of the first envelope is 1/2*a+1/2*2a=3/2*a.
In conclusion, you do not have to change your mind if you choose one envelope.
The key point is that the " x " in the chosen envelope is a random variable, the x in 2x or x/2 for the amount of other envelope is two different value .