Question

In a race between Achilles and tortoise, people assigned probability to Achilles winning and tortoise winning. These probability pairs are listed below. How many of these pairs satisfy the axiomatic approach, assuming only two possible results are tortoise wins and Achilles wins.

• A. $(\frac{1}{2},\frac{1}{2})$
• B. $(\frac{-1}{2},1)$
• C. $(\frac{3}{4},\frac{1}{2})$
• D. $(\frac{-1}{2},\frac{3}{2})$
• E. $(1,0)$

Answer 1

Answer: (A), (E)

$(1,0)$

The probability of sample space is 1. It means that whatever is the outcome, it will be a part of the sample space S. Also, the probability of an event can’t be less than 0 and can’t be greater than 1. It means P(A) belongs to the set [0,1] for any event.
In our problem, the sample space is {Tortoise Winning, Achilles Winning} or {T,A}.
Now, T and A are given in options as pairs. They should satisfy the above condition we mentioned. Only options A and E satisfy the above condition.
For $(\frac{-1}{2},1)$, T is a negative number = $-\frac{1}{2}$. It can’t be negative
For $(\frac{3}{4},\frac{1}{2})$, the sum of the probability is greater than 1. It means that the probability of sample space is greater than 1, which can’t be.
For $(\frac{-1}{2},\frac{3}{2})$, one of the probabilities is negative