Steven G
Elite Member
- Joined
- Dec 30, 2014
- Messages
- 14,561
Let's assume that when we throw a dart at a dart board we will hit the board and where we hit will be random.
Let pi be a point on the board for i in the Naturals
Now P(hit p1) = 0 or P(hit p1) = a where 0 < a < 1
Case 1: P(hit p1) = 0
\(\displaystyle \lim_{n\to \infty}(\sum_{i=0}^n P(p_i)) \ \lim_{n\to \infty}(\sum_{i=0}^n 0) = \lim_{n\to \infty}(n*0) = \lim_{n\to \infty}(0) = 0 \neq1\)
Case 2: P(hit p1) = a where 0 < a < 1
\(\displaystyle \lim_{n\to \infty}(\sum_{i=0}^n P(p_i)) \ \lim_{n\to \infty}(\sum_{i=0}^n a) = \lim_{n\to \infty}(n*a) = \infty\neq1\)
I know that \(\displaystyle P(hit \ p_1) = 0\) but I am not clear where my errors are above.
Let pi be a point on the board for i in the Naturals
Now P(hit p1) = 0 or P(hit p1) = a where 0 < a < 1
Case 1: P(hit p1) = 0
\(\displaystyle \lim_{n\to \infty}(\sum_{i=0}^n P(p_i)) \ \lim_{n\to \infty}(\sum_{i=0}^n 0) = \lim_{n\to \infty}(n*0) = \lim_{n\to \infty}(0) = 0 \neq1\)
Case 2: P(hit p1) = a where 0 < a < 1
\(\displaystyle \lim_{n\to \infty}(\sum_{i=0}^n P(p_i)) \ \lim_{n\to \infty}(\sum_{i=0}^n a) = \lim_{n\to \infty}(n*a) = \infty\neq1\)
I know that \(\displaystyle P(hit \ p_1) = 0\) but I am not clear where my errors are above.