91影视

From the intervals$[0,1]$, there are an unlimited number of possibilities.

$a_1,a_2,a_3,鈥�$

prove:

$\underset{k=1}{\overset{鈭�}{鈭�}}\left[a_k\underset{i=1}{\overset{k-1}{鈭�}}\left(1-a_i\right)\right]+\underset{i=1}{\overset{鈭�}{鈭�}}\left(1-a_i\right)=1$

A series of separate experiments is repeated an endless number of times.

Each time the chances of success is denoted by $a_i鈭�[0,1]$, the probability of success is denoted by $i=1,2,3,$...

Because all of these events are independently exclusive, the first success occurs in the $k$-th experiment with probability $1$.

For $k=1,2,3,...$or all experiments, the first success occurs in the $k$-th experiment with probability $1$

$1=\underset{k=1}{\overset{鈭�}{鈭�}}P$(the first success in $k$-the experiment) $+$$P$(no successes)

The first success in the $k$-th experiment occurs when the first $k-1$experiments fail with probability of $\left(1-a_i\right),i鈮�k,$and the $k$-th experiment succeeds, yielding independence.

$P$(the first success in $k$-th experiment)$=a_k\underset{i=1}{\overset{k-1}{鈭�}}\left(1-a_i\right)$

All tests are independent, and the likelihood that $i$will fail is $\left(1-a_i\right)$, hence the probability that almost all experiments will fail is $\left(1-a_i\right)$.

$P$(no successes) $=\underset{i=1}{\overset{鈭�}{鈭�}}\left(1-a_i\right)$

Substituting the values,

$\underset{k=1}{\overset{鈭�}{鈭�}}\left[a_k\underset{i=1}{\overset{k-1}{鈭�}}\left(1-a_i\right)\right]+\underset{i=1}{\overset{鈭�}{鈭�}}\left(1-a_i\right)=1$