Challenge Level

Here are some statistical and probabilistic statements. However reasonable they might sound, they CANNOT be completely true, or might even be completely false. They are, at best, an approximation, although the approximation might be very good. Why?

- I toss a coin. The chance of landing heads is $1$ minus the chance of landing tails.
- The energy of a randomly selected atom in a box is normally distributed with some mean and variance.
- My pdf is a semi-circle of radius 1.
- For any piece of uranium, half of the mass will decay every time the half-life elapses.
- I record the time of day at which an event occurs, with the knowledge that the event is just as likely to occur at any time as any other. The time measured will be a $U(0,1)$ random variable.
- An influenza pandemic is just as likely to occur one year as any other.
- The arithmetic mean of the number of children per family in a certain population is $a$. Therefore, the arithmetic mean of the number of siblings each child has is $a-1$.
- The expected age of death of a randomly selected $40$ year old is the same as the expected age of death of a randomly selected $35$ year old.

Can you invent your own statistical statements which sound plausible but must be false?