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# S2 Given that a bulb has already lasted 950 hours, what is prob will last 1050 hrs watch

1. Question is: Lightbulb lifetime mean = 1000 hours, sd = 110 hours. Asked to find probability that bulbs will exceed 950 hours and 1050 hours.

exceed 950 hours, p = 0.6754

exceed 1050 hours, p = 0.3246

The part of question I am stuck on is:

Given that a bulb has already lasted 950 hours, what is the probability that it will last a further 100 hours?

This looks like a conditional probability question. So,

A = event that bulb lasts 950 hours

B = event that bulb lasts 1050 hours.

P(B|A) = P(A AND B) / P(A) = 0.6754

P(A) on top and bottom cancel out so answer is P(B)

But this is not the correct answer. Correct answer is 0.481 which is P(B) / P(A). But why?

Can someone please explain?

Is it because P(A AND B) in this case = P(B) ???

This is because the probability of lasting x hours AND x+100 hours will be probability of it lasting x+100 hours. Is there a name for this in probability?

So

P(A|B) which normally equals P(A AND B) / P(B)

equals P(A)/P(B)

Its easy to just pull out standard formulae - but harder to think it through.

In the above if you have P(A) and P(B) and you want to find the probability of both events happening it is tempting just to do P(A) . P(B). But I am thinking this is not appropriate in this scenario? So what are the conditions where you must be careful - and as in this case P(A AND B) = P(B). Or is my understanding wrong?
2. (Original post by acomber)
Question is: Lightbulb lifetime mean = 1000 hours, sd = 110 hours. Asked to find probability that bulbs will exceed 950 hours and 1050 hours.

exceed 950 hours, p = 0.6754

exceed 1050 hours, p = 0.3246

The part of question I am stuck on is:

Given that a bulb has already lasted 950 hours, what is the probability that it will last a further 100 hours?

This looks like a conditional probability question. So,

A = event that bulb lasts 950 hours

B = event that bulb lasts 1050 hours.

P(B|A) = P(A AND B) / P(A) = 0.6754

P(A) on top and bottom cancel out so answer is P(B)

But this is not the correct answer. Correct answer is 0.481 which is P(B) / P(A). But why?

Can someone please explain?

Is it because P(A AND B) in this case = P(B) ???

This is because the probability of lasting x hours AND x+100 hours will be probability of it lasting x+100 hours. Is there a name for this in probability?

So

P(A|B) which normally equals P(A AND B) / P(B)

equals P(A)/P(B)

Its easy to just pull out standard formulae - but harder to think it through.

In the above if you have P(A) and P(B) and you want to find the probability of both events happening it is tempting just to do P(A) . P(B). But I am thinking this is not appropriate in this scenario? So what are the conditions where you must be careful - and as in this case P(A AND B) = P(B). Or is my understanding wrong?
Tbh I hate the conditional probability formula

In this case think in terms of common sense ... If you had 10000 bulbs, 6754 of them will last more than 950 hours ... Of those 3246 will last more than 1050

So you have 3246 out of 6754
3. (Original post by TenOfThem)
Tbh I hate the conditional probability formula

In this case think in terms of common sense ... If you had 10000 bulbs, 6754 of them will last more than 950 hours ... Of those 3246 will last more than 1050

So you have 3246 out of 6754
Thanks.

I think it is safer to think it through when your thinking is fuzzy rather than trotting out standard formulas. As you say, use common sense.

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