do homework 3 #1,3,4,6,9

Homework 3
Question 1: A traditional three digit telephone area code is constructed as follows: The first
digit is from the set {2, 3, 4, 5, 6, 7, 8, 9}, the second is either 0 or 1, the last is from the set
{1, 2, 3, 4, 5, 6, 7, 8, 9}. (a) How many area codes like this are possible? (b) How many such
area codes start with 5?
Solution: Given, the digit that can come at first place is {2, 3, 4, 5, 6, 7, 8, 9}; and the second
digit must be either 0 or 1 and third digit must be from the set {1, 2, 3, 4, 5, 6, 7, 8, 9}
a) Thus, the total number of codes that are possible are:
= 8 x 2 x 9 = 144 codes
This is because only 8 digits can occupy first place, only two can occupy second and 9
can occupy third place in the code.
b) The total number of codes that are possible when the code must start with 5 are:
= 1 x 2 x 9 = 18 codes
Similarly, here only one digit i.e. 5 can occupy first place, two can occupy second place
and nine digits can occupy third place.
Question 3: Seven different gifts are distributed among 10 children so that no child gets
more than one gift. How many different outcomes are possible?
Solution: Now, there are seven gifts which are to be distributed within 10 children. Since no
child gets more than one gifts for the first prize there are 10 choices, for the second prize there
are 9 choices, for the third prize there are 8 choices,…,there are 4 choices for the seventh prize
and all of these choices are independent.
Therefore, the different outcomes that are possible are:
=
10
P7 = 10 x 9 x 8 x 7 x 6 x 5 x 4 = 604800
Question 4: In a certain jurisdiction, it takes at least 9 votes of a 12 member jury to get a
conviction. Assume that
1) 65% of all defendants are guilty;
2) The probability that a juror will convict an innocent is 0.1;
3) The probability that a juror will acquit a guilty is 0.2;
4) Each juror votes independently of the rest of the panel;
Compute the probabilities of the following events: (a) the panel renders a correct decision;
(b) the defendant is convicted.
Solution: Let C be the event that the jury reaches a correct decision, K be the event that the
defendant is convicted, and G be the event...