1. ## 2 Probability question

hello everyone

i am having problem with 2 question :
1) say i don't know the letters a-z and i need to create the word
" Probability" from the letters "P-r-o-b-a-b-i-l-i-t-y " . ( i have exactly 11 places )

2) i have 4 kids with different names but i don't know the names of either of them. what is the prob that i wont call either of them in the right name?

mormor83

2. ## Permutations with repeated items; derangements

Hello mormor83

Welcome to Math Help Forum!
Originally Posted by mormor83
hello everyone

i am having problem with 2 question :
1) say i don't know the letters a-z and i need to create the word
" Probability" from the letters "P-r-o-b-a-b-i-l-i-t-y " . ( i have exactly 11 places )

2) i have 4 kids with different names but i don't know the names of either of them. what is the prob that i wont call either of them in the right name?

mormor83
1) I think the question means: The letters "p-r-o-b-a-b-i-l-i-t-y" are arranged in a random order. What is the probability that they spell the word "probability"?

If this is what the question means, then you need to know the number of permutations (arrangements) of $\displaystyle 11$ items that contain $\displaystyle 2$ items repeated of the first kind ('b') and $\displaystyle 2$ items repeated of the second kind ('i'). This is
$\displaystyle \frac{11!}{2!2!}$

Since just one of these arrangements is the correct one, the probability that this arrangement occurs at random is
$\displaystyle \frac{2!2!}{11!}$
2) Again, if I may re-word the question: the names of 4 children are randomly selected by each child. What is the probability that none of the children selects their own name?

Such a selection - where no item is chosen in its 'natural' place - is called a derangement. With $\displaystyle 4$ items, there are $\displaystyle 9$ derangements. (See, for example, just here.)

It doesn't take long to list all the possibilities with 4 items, but if you want the formula for $\displaystyle d_n$, the number of derangements of $\displaystyle n$ items, it is in the form of a recurrence relation:
$\displaystyle d_n=nd_{n-1}+(-1)^n, n\ge 1,$ with $\displaystyle d_0$ defined as $\displaystyle d_0=1$
You'll see that this gives:
$\displaystyle d_1=0$
$\displaystyle d_2=1$
$\displaystyle d_3=2$
$\displaystyle d_4=9$
$\displaystyle d_5=44$
and so on.

Since the number of arrangements of $\displaystyle 4$ items is $\displaystyle 4!\,(=24)$, the probability that one chosen at random is a derangement is clearly $\displaystyle \frac{9}{24}=\frac38$

thanks for the replay.I appreciate it.

1) got it .thank u
2) i have a little problem with the re-write:
if i have 4 kids. each one has its own name. but i know the name of all 4 but i don't know whose name belong to who.
the question is: if i call them what is the prob that i will miss every one of them ( be wrong every time x4 )

4. Hello mormor83
Originally Posted by mormor83
...2) i have a little problem with the re-write:
if i have 4 kids. each one has its own name. but i know the name of all 4 but i don't know whose name belong to who.
the question is: if i call them what is the prob that i will miss every one of them ( be wrong every time x4 )
Isn't that what I said? The four names are the correct names of the children, but none of them is allocated to the right child.

For instance, if the children are A, B, C and D, one possible order could be B, C, D, A; another could be B, A, D, C. None of the letters is in its 'natural' place. That's a derangement.

so that will give 9/27 possibles ways.

thank u very much for the replay.

mormor83

6. Hello mormor83
Originally Posted by mormor83
You mean $\displaystyle \frac{9}{24}$.