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I'm stucked in two problems:

  1. Suppose that a 1-meter glass rod falls onto the floor and is broken into two pieces. Assume that the breaking point is uniform between 0 and 1 meter. What is the probability that the longer piece has length at least 0.6 meter?

I have solved this one but I don't think it's correct Please look at the picture. I draw a line x+y=1 and then find the area of where x>=6. And then solve the area of the trangle which is 1/2*0.4*0.4=0.08

  1. I solved i) but have no idea for ii). Please see picture attached. Question image

Thank you so much ahead!

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  • $\begingroup$ Hint for the first one: imagine (hypothetically) splitting the rod into $5$ sections of $0.2$ meters each. In how many out of these $5$ sections can the break occur and result in the longer piece being at least $0.6$ meters? $\endgroup$ – wgrenard Oct 1 '16 at 22:39
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Suppose $Z$ is the distance between the breaking point and one end point of your rod. Then Z~U(0,1)

So P(The longer piece has length at least 0.6 meter)=

$P(Z>0.6$ or $Z<0.4)=P(Z>0.6)+P(Z<0.4)=(1-0.6)+(0.4)=0.8$, that's the answer of you question.

Notice that you cannot use two dimension plot to help you solve this problem because X and Y are correlated, there is no way to solve it by this kind of plot.

5ii:

You should first show that in order to prove 5ii, you need to prove:

$P(AB)+P(BC)-P(AC)<P(B)$

Hint:using 5i to expand every item.

Then you should know $P(A\cup B)=P(A)+P(B)-P(AB)$

Try to expand $P\{(AB)\cup (BC)\}$ and prove $P(AB)+P(BC)-P(AC)<P\{(AB)\cup (BC)\}$, then prove $P\{(AB)\cup (BC)\}<P(B)$, you will get the answer.

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  • $\begingroup$ Hi Jeff, thank you for your solution. It's very clear and I understand it now. Thank you so much! $\endgroup$ – MathLearner1 Oct 1 '16 at 22:53
  • $\begingroup$ Do you have any thoughts for 5(ii)? $\endgroup$ – MathLearner1 Oct 1 '16 at 22:59
  • $\begingroup$ See the update @MathLearner1 $\endgroup$ – Jeff Oct 1 '16 at 23:15
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The lengths of the two pieces are not jointly uniformly distributed (they are obviously not independent!) so you cannot use the "area" approach.

Just consider where you could break the rod to obtain a piece that is at least $0.6$ meters. Hint: the breaking point just needs to be within $0.4$ meters of one of the ends of the rod.

$0.8$.


It may help to draw a triple venn diagram.

Hint: show $(A \triangle B) \cup (B\triangle C) = (A \cup B \cup C) - (A \cap B \cap C)$ and note that this set contains $A \triangle C$.

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  • $\begingroup$ Thank you for your response! It helps but I'm still confused with 5 (ii). For the 5 (i), I have already worked it out. Do you have any thoughts for 5(ii)? $\endgroup$ – MathLearner1 Oct 1 '16 at 22:59
  • $\begingroup$ @MathLearner1 I already wrote my hint for part (ii) $\endgroup$ – angryavian Oct 1 '16 at 23:23

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