00:01
In this question it is given that we have to let our value for sequence we are non -decreasing sequence of event.
00:19
Sequence of event and we need to show that limit n tends to infinity for b -c limit is equal to probability of getting a value for n is equal to 1 to infinity for c f so now let's see the solution of this part so first of all for part a we will define here b1 as equal to c1 b2 will be equal to c2 by c1 and here value of b3 is liquid to c3 by c2 so we can see this by the diagram we have three sequence circles here.
01:13
So this one refers to b1, then outer circle here is refers to b2, then next circle this one is referring to b -3.
01:28
So thereafter.
01:31
Okay, so this sequence will go up to any term.
01:34
So b -n is equal to cn by cn minus 5, where n is greater than or equal to.
01:44
So therefore, this implies as a value for n is equal to 1 to infinity of f f f fn will be equal to the n is equal to n is equal to 1 to infinity of c n and when we solve this this will be a spiel unity of n is equal to 1 infinity of cn is equal to priority of p bn is equal to n is equal to 1 to infinity versus infinity for bn will be equal to summation of n is equal to 1 to infinity for p b n and here b n level b n where the limits are n minus 1 to infinity is the disjoint event.
02:47
Joint event.
02:51
So now, our value will be, this implies us, the beauty of, meaning of n is equal to 1 to infinity for cn is equal to limit.
03:04
N tends to infinity for summation of n is equal to 1 to n, p, b at k.
03:14
Here we have our first equation and now we can say by the equation 1, we can find bk is equal to c k by c k minus 1.
03:30
So here the value of c k will be equal to c k minus 1.
03:36
2 meaning of c k by c k minus 1.
03:42
And this implies as probability of getting given c k is equal to priority of getting c k minus 1 plus probability of getting c k divided by c k minus 1.
03:58
So here we will have next step as probability of getting b k is equal to probability of c k minus probability of c k minus 1.
04:11
So then after we will see therefore by equation 1 are the value as probability of union of n is equal to 1 to infinity for m is equal to limit n tends to infinity for the values p c1 added by p c2 added by minus p c1 and so on up to p cn minus p cn minus c n minus one so the sum will be as our answer for union n is equal to one to infinity of c n will be equal to limit n tends to infinity for p c n so, hence we have proved the part b and the proof.
05:27
This is our solution for part h.
05:31
Now, in the part b, here we will consider our value for an f term as n is equal to 1 in 20, be the increasing sequence...