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# Probability (Bernoulli & Expectation)...

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1. Probability (Bernoulli & Expectation)...
A group of men exchange their gloves randomly (altogether there are 20 gloves).

Let be the random variable which attains the value 1 if the man got at least one of his gloves back, and it attains the value 0 otherwise.

Set and .

Show that and that .
number of men who got at least one of their gloves back.

For each , consider,

if the man got at least one of his gloves back.
______ if he didn't.

.

Each has the Bernoulli distribution of parameter,

How would I go about working out or ?
Last edited by RamocitoMorales; 03-05-2012 at 20:07.
2. Re: Probability (Bernoulli & Expectation)...
(Original post by RamocitoMorales)
?
Assuming the gloves are now randomly scattered.

For the i'th man, what's the probability that he doesn't have his first glove? (How many of the 20 slots can it be in?)

Similarly the probability for his second glove, and multiply.
Last edited by ghostwalker; 03-05-2012 at 20:19.
3. Re: Probability (Bernoulli & Expectation)...
(Original post by ghostwalker)
Assuming the gloves are now randomly scattered.

For the i'th man, what's the probability that he doesn't have his first glove? (How many of the 20 slots can it be in?)

Similarly the probability for his second glove, and multiply.
Would that not just be,

What have I done wrong (if anything), as the calculated answer is not equal to the solution I am after.

EDIT: gives me what I'm after. I apologise, and thank you.
Last edited by RamocitoMorales; 03-05-2012 at 20:32.
4. Re: Probability (Bernoulli & Expectation)...
(Original post by RamocitoMorales)
Would that not just be,

What have I done wrong (if anything), as the calculated answer is not equal to the solution I am after.
Once you've chosen a position for the first glove, there are only 19 positions remaining for the second.
5. Re: Probability (Bernoulli & Expectation)...
(Original post by ghostwalker)
Once you've chosen a position for the first glove, there are only 19 positions remaining for the second.
Yes, thank you, I've corrected that error.
6. Re: Probability (Bernoulli & Expectation)...
What if I now wanted to work out:

Let be the total number of men who received at least one of their gloves. Find .
By the linearity of expectation:

, where is the event that no man receives a single one of their gloves back and is the event where all ten men receive at least one of their gloves back.

Help?
7. Re: Probability (Bernoulli & Expectation)...
Anyone?

I am guessing that I cannot just do .
8. Re: Probability (Bernoulli & Expectation)...
(Original post by RamocitoMorales)
What if I now wanted to work out:

By the linearity of expectation:

, where is the event that no man receives a single one of their gloves back and is the event where all ten men receive at least one of their gloves back.

Help?
Puzzled.

You seem to have changed the definition of .

With your original definition, it's straight forward, but with this new one, I don't know.

Edit:
Thinking about it, your new definition, doesn't make sense.

So, I presume you meant your previous definition, in which case,

Since n=10, your index runs 0 to 9, or 1 to 10, but not 0 to 10.

And then you can just add.
Last edited by ghostwalker; 04-05-2012 at 12:25.

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Last updated: May 4, 2012
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