Question

What is the probability of randomly choosing a permutation of the ten digits 0, 1, 2, . . ., 9 in which an odd digit is in the first position and 1, 2, 3, 4, or 5 is in the last position?

Answer #1

The 10 decimal digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are arranged
in a uniformly random permutation. We denote by a the integer
formed in base 10 by the first five positions in this permutation
and by b the integer formed in base 10 by the last five positions
in this permutation (either a or b may begin with 0 which in such a
case is ignored). For example, if the random permutation is
8621705394 then...

What is the probability of these events when we randomly select
a permutation of the 26 lowercase letters of the English alphabet?
(a) a is the first letter of the permutation and z is the last
letter. (b) The 5 vowels are not all separated. (c) The first 3
letters are all together in the permutation (in some order) or the
last 4 letters are all together in the permutation (in some order).
(d) The first 13 letters of the...

The 10 decimal digits, 0,1,2,3,4,5,6,7,8,9 are arranged in a
uniformly random permutation. We denote by "a" the integer formed
in base 10 by the first five positions in this permutation and by
"b" the integer formed in base 10 by the last five positions in
this permutation (either a or b may begin with 0 which in such a
case is ignored). For example, if the random permutation is
8621705394 then a= 86217 and b= 5394. Consider the probability
space...

We have the
digits{2,3,⋯7} .
If the digits are
randomly arranged in a row, find the probability that the first
digit
is 3 and the last
digit is 2.

1) a. Find the probability of choosing, with replacement, a 5
and then an ace out of a standard deck of cards.
b. Raffle tickets for a trip to Miami are assigned three digit
numbers using the digits 0-9. Find the probability that the number
on the winning ticket is either even or less than 300?

(a) How many three-digit numbers can be formed from the digits
0 comma 1 comma 2 comma 3 comma 4 comma 5 comma 6 comma 7 comma 8
comma and 9
if each digit can be used only once?
(b) How many of these are odd numbers?
(c) How many are greater than 440?

A large collection of one-digit random numbers should have
about 50% odd and 50% even digits\ because five of the ten digits
are odd (1, 3, 5, 7, and 9) and five are even (0, 2, 4, 6,
and 8).A. .Find the proportion of odd-numbered digits in the
following lines from a random number table. Count carefully:
9 9 2 8 9 7 6 5 8 3 0 8 6 0 2
3 7 8 4 9 9 2 8...

10. Three-digit numbers are to be made from the 10 digits 0, 1,
2, …, 9. (Assume that the first digit cannot be 0.)
a. How many numbers can be made if repetitions are not allowed
and the number must be divisible by 10?
b. How many numbers can be made if repetitions are allowed and
the number must be even?
c. How many numbers can be made if repetitions are not allowed
and the number must be odd and...

A three-digit numeral is formed by selecting digits from {0, 3,
7, 9} without repetition.
Find the probability that the number formed is:
More than 300
Less than 750
Contains only odd digits
An even number.

It is a striking fact that the first digits of numbers in
legitimate records often follow a distribution known as
Benford's Law, shown below.
First digit
1
2
3
4
5
6
7
8
9
Proportion
0.285
0.186
0.126
0.084
0.069
0.067
0.034
0.039
0.110
Fake records usually have fewer first digits 1, 2, and 3. What
is the approximate probability, if Benford's Law holds, that among
1153 randomly chosen invoices there are no more than 670 in amounts
with...

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