Stat Help.......Please?

Question

A playful secretary typed 8 different bussiness letters, addressed the eight corresponding envelopes, and then distrubuted the letter among the envelopes at random.
(a) Probability that no one will get the right letter
(b) At least one person will get the right letter

2)

i am confused with these answers. Is it a combination problem? what is the sample space here?

Answer 1

luv phy is doing the calculations as if each letter could be placed in any one of the 8 envelopes, even after some of the envelopes have been used for other letters. In other words, he is treating the placing of the letters as 8 independent events. But they are not independent, as each event is affected by what has happened before.

The probability that no one will get the write letter is equal to the number of ways that can happen, divided by the total number of ways that the 8 letters could possibly be arranged. The second number (the denominator) is easy: it's 8! = 8x7x6x5x4x3x2x1 = 40,320.

To get the numerator, you have to do a calculation. There may be a standard formula, but one rule that works is this:
The number of ways that n envelopes can all contain the wrong letter (call this number W(n) ... "W" for "wrong") is
(n-1) x (W(n-1) + W(n-2)).
This applies when n is 3 or more.
For n = 1, W(1) = 0 (i.e., you can't screw up if there's just one letter and one envelope).
For n = 2, W(2) = 1 (i.e., there's only one way to screw up 2 letters: you get them reversed).

So, applying the formula,
W(3) = (3-1) x (1 + 0) = 2
W(4) = (4-1) x (2 + 1) = 9
W(5) = (5-1) x (9 + 2) = 44
W(6) = (6-1) x (44+ 9) = 265

I'll let you work out the values for W(7) and W(8).
Then you need to divide W(8) by 40,320 to get the probability requested in (a) of the problem.

Once you solve (a), then (b) is simple. It's just 1 - (a). Either no one gets the right letter, or someone (at least one person) gets the right letter. So the answers to (a) and (b) have to add up to 1.

Good luck!

Answer 2

8 letters to 8 recipients forms an 8X8 matrix of possible outcomes. Each of 8 rows is a letter and each of 8 columns is a recipient. Thus, there are 8X8 = 64 possibilities (possible combinations of letter-recipient).

What you want to know is the probability that letter 1 goes to recipient 1 as intended, and so forth for the 8 letters and recipients. Let's call this P(1,1), P(2,2), etc. P(8,8) meaning the probability that letter 1 goes to recipient 1, 2 goes to 2, etc.

The probability that no one will get the right letter is P(p,q); where p<>q. In the 8X8 matrix, the only way no one will get the right letter is for all the outcomes to fall off the diagonal where p=q; e.g., P(1,1) and P(7,7) are on the diagonal, but P(2,3) and P(6,1) are not.

(a) There are eight outcomes on the diagonal; so P(p,p) = n(p,p)/N = 8/64 = 1/8, which is the probability that everyone will get their intended letter. So the probability no one will get his letter is P(p<>q) = 1 - P(p,p) = 1 - 1/8 = 7/8.

(b) The probability that "at least" one recipient will get the right letter can be written as P(n(p,p)>=1) = 1 - P(n(p,p) < 1); where n(p,p) < 1 means the number of correct matchups is less than one; but that is zero matchups because zero is the only feasible number less than one in this case. n(p,p) < 1 or, equivalently, n(p,p) = 0 occurs only off diagonal so we have n(p,q) where p<>q; so there are n(p,q) = N - n(p,p) = 64 - 8 = 56 possible ways to get no matchup at all.

This means that P(n(p,p)>=1) = 1 - P(n(p,p) < 1) = 1 - n(p,q)/N = 1 - 56/64 = 1/8, which is the probability at least one recipient will get the correct letter.

This result is intuitively pleasing because it says that getting at least one right is the logical opposite of getting them all wrong. That is, as the likelihood of getting them all wrong goes up the chances of getting at least one right go down.

PS: You have a right to be confused, the answers are way off mark. Your sample space, which is actually the population is N = 64 possible outcomes. There are only n(p,p) = 8 possible correct matchups. This means there are 64 - 8 = 56 ways to not match a letter to its proper recipient.

Answer 3

Probability of one letter getting an incorrect envelope is 7/8
So the probability that none will get their correct envelope is (7/8)^8 = 5764801 / 16777216

similarly, 1/8 (probability of 1 correct) x (7/8)^7 (probability of 7 incorrect) = 823543 / 16777216.

Hope this helps=)

Answer 4

The person above is correct for part A. However, he calculated the probability of exactly one person getting the right letter for b. To calculate the probability for at least one person, you can just do this:
1 - answer from part a.
This is because the chances of at least one person getting the correct letter is the opposite of the chances of no one getting a correct letter.