The probability that:
a fair coin gives a head when tossed is 1/2
a fair die shows a 6 is 1/6
an ace is drawn from a deck of cards is
4/52=1/13=0.0769
In each case all these values coincide with intuition.
I play a guessing game with Jack and Jill. Jack selects a fair coin, tosses it, and conceals the up turned face:
He asks me what is the probability that the coin is showing a
head.
I answer 1/2, by symmetry .
Jack shows me the coin.
It shows a head.
He asks me what is the probability that the coin is showing a head.
I answer that the probability is 1.
Jack shows the result of a new toss to Jill while concealing it
from me.
He asks me what is the probability that the coin is showing a head.
How should I now reply this time?
This example poses questions concerning the nature of probability. Is probability a physical property of the coin or is it somehow also determined by the experiment or by the experimenters and the observers?
Three indistinguishable purses each contain two coins.
One purse contains two gold coins, another contains two silver coins
and the third contains one gold coin and one silver coin.
GG GS SS
A purse is selected at random and at random a coin is selected from
it.
It turns out to be gold.
What is the probability that the other coin in that purse is also
gold?
The first gold coin can have been selected from either the purse containing two gold coins or from the purse containing a gold and a silver coin.
Intuition might suggest that the probability of the second gold must be as only one of these two events leads to another gold coin.
We shall later see that both this argument and the answer are false.
By giving simple motivating examples the course will emphasise how random variables and probabilities occur in practice. Naturally, as this is a first course in probability, these examples are mathematically simple. But even with this restriction, we will see problems of real practical relevance in addition to simple coin-tossing.