Probability: Conditional and Bayes

chineseballer06 · 2011-02-15 18:18:21

Hi everyone, I need some help with a couple problems and appreciate any help that I can get.

The first problem is: "A die is rolled to yield a number between 1 and 6, and then a coin is tossed that many times. What is the probability that heads will not appear?

The answer I got is 1/6(.5+.5^2+.5^3+.5^4+.5^5+.5^6) but I don't know if that is the right way to go about it.

The second problem I am just completely lost and its: "It is believed that a sought-after wreck will be in a certain sea area with probability p = 0.4. A search in that area will detect the wreck with probability d = 0.9 if it is there. What is the revised probability of the wreck being in the area when the area is searched and no wreck is found?

Any help is greatly appreciated!

bobbym · 2011-02-15 20:33:13

Hi;

The first answer is correct.

gAr · 2011-02-15 23:05:08

Hi chineseballer06,

Your first answer is correct.

P(Wreck not being found in the area) = 0.64
So, can we revise the probability of the wreck being in the area to 0.36, when no wreck is found? I'm not sure.

chineseballer06 · 2011-02-16 05:31:51

Hi bobbym and gAr,

How did you find P(Wreck not being found in the area) ?

gAr · 2011-02-16 05:52:30

Hi chineseballer06,

P(Wreck not being found in the area) = P(Wreck in the area)*P(wreck is not found) + P(Wreck not in the area) = 0.4*0.1 + 0.6 = 0.64

Bob · 2011-02-16 06:25:16

hi chineseballer06 and gAr

See my tree diagram below.

Let's suppose there in the multiverse there are 100 identical seascape universes.

Identical except that the wreck is only there in 40 of them.

In 60 universes there is no wreck.

There chance of finding the wreck if it's there is 9/10 or 36/40, so there are 36 universes where the wreck exists and is found.

There are 4 universes where the wreck is there but not found.

There are 0 universes where the wreck isn't there but has been found. (I've included this case just for completeness; in some problems it won't be zero.)

There are 60 universes where the wreck isn't there and it hasn't been found.

But now the search is made and we are told the wreck hasn't been found.

So only the 64 universes are left in our multiverse.

Out of those only 4 have the wreck.

So the probability that the wreck is now in the seascape is reduced to 4 / 64.

Bob

Last edited by Bob (2011-02-16 09:39:53)

chineseballer06 · 2011-02-16 12:21:07

Oh okay!! Thanks that makes a lot of sense. Just curious, do you know how to do this using the conditional probability formula or Bayes' Formula?

gAr · 2011-02-16 15:08:55

Hi bob,

Thanks for explaining. The word "revise" confused me.
I too use tree diagrams for such problems, problem becomes easier when we can visualize it.

Hi chineseballer06,

Let W = Wreck is in the area ; N = Wreck is not found
We require:

chineseballer06 · 2011-02-16 16:03:21

Yeah that does help to see it, and thanks for the conditional probability example! I appreciate the help everyone!!

gAr · 2011-02-16 16:26:18

Hi chineseballer06,

You're welcome.

Bob · 2011-02-16 22:57:21

hi chineseballer06 and gAr

You're welcome.

Thanks from me for the formula. As you will learn from my posts, I'm hopeless at formulas; I much prefer going back to first principles. This question was a gift for me as I could make a diagram too. In exams, my scrap paper was always full of scribbled working as I re-proved the formulas I wanted. Slows you a bit, but it makes it easier when the question says " Prove that ......"

Bob

gAr · 2011-02-16 23:21:47

Hi bob,

I remembered this formula just to write in the exams, but to solve I used tree diagrams.
Most of the problems related to conditional probability are best solved using tree diagrams.

Bob · 2011-02-17 00:54:09

hi gAr

That's the same with me except I'd have to use the diagram to remind me what the formula is.

What a strange old world when we have to use the easier method to quote the 'official' method just to keep the examiners happy.

There is an apocryphal story that does the rounds every year at undergraduate exam time about a Cambridge student who answered a maths question in his finals in just two lines! His professors had expected their best student to take his exams a little more seriously than that! But the more they thought about his answer the more they came to realise that his answer stood up and was just a different, and somewhat shorter, way of doing the question. So they awarded him a 'first'!

While on the subject of apocryphal stories, there is another about a philosopy student who was given the question: "Is this a question", to which he gave the response: "If that is a question, then this is an answer!"

Bob

Last edited by Bob (2011-02-17 00:55:46)

Math Is Fun Forum

#1 2011-02-15 18:18:21

Probability: Conditional and Bayes

#2 2011-02-15 20:33:13

Re: Probability: Conditional and Bayes

#3 2011-02-15 23:05:08

Re: Probability: Conditional and Bayes

#4 2011-02-16 05:31:51

Re: Probability: Conditional and Bayes

#5 2011-02-16 05:52:30

Re: Probability: Conditional and Bayes

#6 2011-02-16 06:25:16

Re: Probability: Conditional and Bayes

#7 2011-02-16 12:21:07

Re: Probability: Conditional and Bayes

#8 2011-02-16 15:08:55

Re: Probability: Conditional and Bayes

#9 2011-02-16 16:03:21

Re: Probability: Conditional and Bayes

#10 2011-02-16 16:26:18

Re: Probability: Conditional and Bayes

#11 2011-02-16 22:57:21

Re: Probability: Conditional and Bayes

#12 2011-02-16 23:21:47

Re: Probability: Conditional and Bayes

#13 2011-02-17 00:54:09

Re: Probability: Conditional and Bayes

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