00:01
In this question we're told that the time lasted by a bowl is exponential with an average of 1 ,000, which means that it has a rate of 1 over 1 ,000.
00:16
So that means that the density of t is going to be 1 over 1 ,000, e to the minus x over 1 ,000, sorry, e to the minus t of 1 ,000, for t greater than 0.
00:38
So first of all, we want to find the probability that both of the lamps bulbs fail within a thousand hours.
00:48
So if we have t1, t2 are independent and identically distributed according to t.
00:54
What's the probability that t1 is less than 1 ,000 hours and t2 is less than 1 ,000 hours as well? well, this is just the probability that t is less than 1 ,000 hours squared because they're independent.
01:08
So this is going to be the integral from 0 to 1 ,000, 1 over 1 ,000, e to the minus t over 1 ,000 d t squared.
01:22
So that's minus e to the minus t over 1 ,000 between 0 and 1 ,000 squared.
01:32
So this is going to be 1 minus e to the minus 1 squared.
01:38
So that gives us 1 minus 1 over e squared.
01:45
Is 0 .3996.
01:56
So that's the probability that they both fail within a thousand hours.
02:00
So t1, t2, less than a thousand.
02:08
Okay, so another lamp has just one bowl.
02:12
And so if we say that now, t1 is the lifetime of bulb one, t2, t2, is the lifetime of bulb 2.
02:33
We want to know what's the probability that they both fail within a thousand hours.
02:36
So what's the probability that t1 plus t2 is less than a thousand? well, that's the probability that t2 is less than a thousand minus t1...