Determine the probability of ##P(x>4)##

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Homework Statement
see attached
Relevant Equations
Knowledge of distributions
1649156003777.png


Find the solution here;

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Ok my interest is on part (b) and (c) only. Let's start with (b),
My take is,
$$\int_4^5 \dfrac{2}{75}x\,dx=\left.[\frac{x^2}{75}]\right|_4^5$$
$$=(0.33333333-0.21333333)+\frac{2}{15}×5$$
$$=0.12+0.6666666666=0.78666666$$
note that at ##f(x)##=##\dfrac{2}{15}##, the probability value will be the area subtended by values of ##x## from ##x=5## to ##10## hence ##10-5=5##...in short, probability ##P(x>4)## at this point will be the area subtended by the straight line within the given domain ##5<x≤10##.
 
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You are confusing the reader with all those decimals. The general rule is
<br /> \mathbb P(X\leqslant a) = \int _{-\infty}^a f(t)\mathrm{d}t,<br />
if ##X## has a probability density ##f##. For b) apply what you know about the probabilities of an event and its opposite event.

If you are looking for feedback, explain your thought process that leads you to a certain computation. The probability you are looking for in c) is that at least one battery fails within the next ##40## hours, because both batteries are required for the device to work.
 
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nuuskur said:
You are confusing the reader with all those decimals.
Yes, you should not use decimals in a problem like this; express your answer as a rational number. And look up the meaning of the word "subtend", it has nothing to do with area.

As to method: you have correctly worked out ## P(x>4) ## by evaluating an integral at 2 points and adding on an area of a rectangle. Can you think of a way to arrive at the same answer simply by calculating the area of a triangle? (Perhaps it would have been better not to ignore part (a).)
 
pbuk said:
Yes, you should not use decimals in a problem like this; express your answer as a rational number. And look up the meaning of the word "subtend", it has nothing to do with area.

As to method: you have correctly worked out ## P(x>4) ## by evaluating an integral at 2 points and adding on an area of a rectangle. Can you think of a way to arrive at the same answer simply by calculating the area of a triangle? (Perhaps it would have been better not to ignore part (a).)
Ok I will look at that...I had already seen how the graph looks like in part (a)...we have a right angle triangle and a rectangle...noted on 'subtended'...
 
chwala said:
Ok I will look at that...I had already seen how the graph looks like in part (a)...we have a right angle triangle and a rectangle...noted on 'subtended'...
OK, so if you divide the area under the graph into two pieces by drawing a vertical line at ## x = 4 ##, you have a triangle with a known base and a height which can be calculated with a trivial substitution. You also have an irregular pentagon. For which piece is it easier to calculate the area?
 
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Thanks @pbuk ...kindly let me look at this later...when I am through with other tasks...
 
Ok we shall have,
##\left[(\dfrac{1}{2}×5×\dfrac{2}{15})-(\dfrac{1}{2}×4×\dfrac{8}{75})\right]+5×\dfrac{2}{15}=\dfrac{59}{75}##
 
chwala said:
Ok we shall have,
##\left[(\dfrac{1}{2}×5×\dfrac{2}{15})-(\dfrac{1}{2}×4×\dfrac{8}{75})\right]+5×\dfrac{2}{15}=\dfrac{59}{75}##
You are still over-complicating this, you should be able to see on inspection that
## P(x \gt 4) = 1 - P(x \le 4) = 1 - \dfrac{1}{2}×4×\dfrac{8}{75} = \dfrac{59}{75}##.
 
pbuk said:
You are still over-complicating this, you should be able to see on inspection that
## P(x \gt 4) = 1 - P(x \le 4) = 1 - \dfrac{1}{2}×4×\dfrac{8}{75} = \dfrac{59}{75}##.
Thanks mate.
 
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Find the solution below from the Markscheme.

Now for part (c), i was not understanding how they came up with using part (b) to realize the solution i.e

1649328188537.png


...then i realized from the question itself indicate...in tens of hours... implying in our case, ##4## × tens of hours = ##40## hours. The steps to final solution is very much clear to me. Cheers guys.
 

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