MA/CS 109 Homework:
Due: In Class, Weds Oct 21
(NOTE: If you type something like
= 247/(45*45*45)
in a Google search box, it will compute the result.)
1 Suppose West Dakota decides that it will issue license plates for its cars that have
two letters (every car gets a different plate):
a. How many possible license plates are there (Hint: There are 26 ways to pick
the first letter and for each choice of the first letter, there are 26 ways to pick
the second letter—use the multiplication principle.)?
b. If all license plates are chosen using the equally likely outcomes model, what is
the probability that the governor will get a license plate that starts with the
letter A?
c. If all license plates are chosen using the equally likely outcomes model, what is
the probability the governor will get a license plate with two of the same
letter?
Answers: a.) Since the first letter can be chosen any of 26 ways and the second
letter can be chosen any of 26 ways no matter what first letter is chosen, there
are 26 x 26 = 676 possible plates.
b.) There is one way to pick the first letter equal to A and 26 ways to pick the
second letter so 26 possible plates that start with A. So the probability of
getting a plate that starts with A is 26/676 = 1/26.
c.) There are 26 plates with 2 of the same letter, so the probability of getting
such a plate is 26/676 = 1/26.
2. Going back to West Dakota in problem 1, bestwordlist.com says there are 124
two letter words in English.
a. If all license plates are chosen using the equally likely outcomes model, what
is the probability that the governor has a license plate that is a 2 letter word?
b. If all license plates are chosen using the equally likely outcomes model, what
is the probability that neither the governor nor the governor’s press
secretary has a 2 letter word as a license plate? (Hint: Remember the
birthdate problem and count carefully.)
Answers: a.) The number of 2 letter word outcomes is 124 so the probability
of getting a plate with a 2 letter word is 124/676 = 0.183…
b.) This is a little trickier. There are 26 x 26=676, but then there are only 675
ways to pick the press secretary’s plate since one plate went to the governor.
Hence there are 676 x 675 ways to pick the plates for both of them. The
number of outcomes where neither has a 2 letter word is
(676-124) x (675-124) the governor doesn’t get a word and so there are 676124 plates to choose from and the press secretary doesn’t get a word either
so there are 675-124 plates to choose from. Hence the probability of neither
getting a word is (676-124) x (675-124)/(676 x 675).
3. Suppose East Dakota decides that it will issue license plates for its cars that
have 6 letters and every car gets a different plate.
a. How many possible license plates are there?
b. If all license plates are chosen using the equally likely outcomes model,
what is the probability that the governor will get a license plate that is the
same letter repeated 6 times?
c. The web page bestwordlist.com says there are 22157 six letter words in the
English language. Again, if all license plates are chosen using the Equally
Likely outcomes model, what is the probability that the governor will get
a license plate that is an English word?
Answer: a.) Again by the multiplication principle, there are
26 x 26 x 26 x 26 x 26 x 26 = 266 (=308,915,776) possible plates.
b.) There are 26 ways for the governor to get a plate of all one letter, so
the probability is 26/ 266 = 1/265 about 0.00000008
c.) The probability of the governor getting a word is 22157/266 which is
about 0.00007.
4. Suppose two candidates, A and B are running for office. A poll is taken of 100
likely voters. Of the 100 voters, 57 say they will vote for A and 43 will vote for B.
Candidate A says “oh goody, I am sure of winning!” Is this correct? Why or why
not?
Answer: No, this is not correct. The margin of error at the 95% confidence level is
1/square root of 100 = 1/10 = .1 or 10%. What can be said is that we are 95%
confident that between 47% a and 67% of the population supports A. So we are
not 95% confident A will win.
5. Suppose two candidates, A and B are running for office. A poll is taken of 1000
likely voters. Of the 1000 voters, 570 say they will vote for A and 430 will vote
for B. Candidate A says “oh goody, I am sure of winning!” Is this correct? Why or
why not?
Answer: No, this is not correct either. The margin of error in this case is
1/square root of 1000 or about .03 or 3% so we are 95% sure that between 54%
and 60% of the population supports candidate A. So we are at least 95%
confident A will win, but not 100% confident.
6. Suppose two candidates, A and B are running for office. A poll is taken of 1000
likely voters. Of the 1000 voters, 570 say they will vote for A and 430 will vote
for B. What can you say to candidate A from this information? Candidate A is
smart, but does not know any statistics…so use language that candidate A can
understand.
Answer: Since we are only looking at a sample (1000 voters out of the entire
population of voters) we can not be sure that the population will vote the same
as the sample. However, we can be 95% confident that you will get more thant
54% of the vote since most samples have are very similar to the population.
7. Suppose you watch someone flip a coin 100 times and they get 72 heads and
28 tails. What can you say about the “fairness” of the coin and/or the person
flipping the coin? Be precise.
Answer: Since we flip 100 times, we have a margin of error of 1/10 = 0.1 or 10%.
Since 72% of the flips came out heads, we are 95% confident that over the long
run between 62% and 82% of the flips will come out heads. So we are at least
95% confident that this is not a fair coin or fair flipping.
8. Hand in the work sheet from class Monday. Did you show at the 95%
confidence level that your two groups had different proportion of “yes” answers
to your question? Why or why not?
Remark: It is unlikely that your results gave 95% confidence of a difference
between the groups because your samples were small.