Download 2014 Q1 Exam Review

Exam Next Class!
Warm Up 10/20
Quiz Question:
Four different numbers are randomly chosen from the set S = {1, 2, 3,
4, 5, …, 10}. X is the second largest the numbers selected. Determine
the probability that X is:
a. 2
b. 7
c.9
Many students students understandable got 0, 1/56, and 1/90 as the
answers to a, b, and c respectively.
1) Explain how they got their answers.
2) What didn’t they take into account?
3) Change their answers so that they are now correct.
Objectives on Exam
5.1a
SWBAT… Understand and analyze concepts of population, sample, random
sample, and frequency distribution of discrete and continuous data.
5.1b
SWBAT… Utilize grouped data: mid-interval values, interval width, upper and
lower boundaries.
5.1c
values.
SWBAT… calculate and interpret mean, variance, and standard deviation
5.2
SWBAT… Concepts of trial, outcome, equally likely outcomes, sample space
(U) and event; the probability of event A as P(A)=n(A)n(U) ; complementary events A and
A’; Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve
problems.
5.3
SWBAT… determine probabilities for combined events, understand and utilize
the formula for P(A⋃B), determine mutually exclusive events.
5.4
SWBAT… calculated conditional probabilities, probabilities for independent
events and use Bayes’ theorem for a maximum of three events.
5.5
SWBAT… Understand concepts of discrete and continuous random variables
and their probability distribution. Define and use probability density functions. Find
expected values (mean), mode, median, variance and standard deviation. Apply random
variables and expected value.
Probability Quiz
1) a. 0
b. 3/14 c. 2/15
2) TBA
3) a. 3/8
b. 1/8
c. ¼
4) a. 11/20 b.i. 2.8 b.ii.
c.i. 9/400 c.ii. 7/40
5) TBA
Probability – Non-Calc
P(X’|Y) = ⅔, P(Y) = ⅚ and P(X’∩Y’) = ∅
Find P(X)
Probability – Non-Calc
The probability that a man will be alive in 25 years
is 3/5, and the probability that his wife will be alive
is 2/3. Determine the probability that in 25 years:
a.
b.
c.
d.
Both will be alive
At least one will be alive
Only the wife will be alive
Rewrite a, b and c in set notation
Probability – Calc
On any day it could rain with 25% chance and
be windy with 36% chance.
a. Draw a tree diagram showing the
possibilities with regard to wind and rain on
a particular day
b. Hence determine the probability that on a
particular day there will be
i. rain and wind
ii. Rain or wind or both
c. What assumptions have you made in your
answers?
Probability - Calc
The students in a school are all vaccinated
against measles. 48% of the students are
males, of whom 16% have an allergic reaction
to the vaccine. 35% of the girls also have an
allergic reaction. Find the probability that a
randomly selected student:
a. Has an allergic reaction
b. is female give that a reaction occurs
Probability – w/ Calc this time
Five different numbers are randomly
selected from the set:
S = {2x| 3 ≤ 2x ≤ 26, x∈Z }
Find the probability that the second largest
number is:
a. 7
b. 22
c. 8
d. 14
Probability – Calc
Using a 52 card pack, a ‘royal flush’ consists
of the 10, J, Q, K, A of one suit. Find the
probability of dealing:
a. A royal flush in any order
b. A royal flush in the order 10, J, Q, K, A
Probability - Calc
Two players, A and B, alternately throw a fair
six–sided dice, with A starting, until one of
them obtains a six. Find the probability that
A obtains the first six.
Probability - Calc
Bradford and Nate are bored in math class and start playing with
the dice and eventually decide to play a game. The rules are
simple they alternately throw a fair six–sided dice, with Bradford
starting, the one that rolls the higher number scores a point.
a. Find the probability that Bradford wins the first point
b. To make things more interesting the play a best of 5 (meaning
first to 3 wins). Find the probability that Nate wins within 5 turns.
c. Why is the “within 5 turns” qualifier necessary? How would
you figure out the true probability of Nate winning? (You do not
need to actually calculate the probability)
Probability – No Calc
In the television show Numb3rs (currently on Netflix) they
referred to Simpson’s paradox, with math genius Prof. Charlie
Eppes explaining it to viewers and the FBI using the following
tables:
1995
1996
1997
Combined
Derek Jeter
12/48
.250
183/582
.314
190/654
.291
285/128
4
.300
David
Justice
104/411
.253
45/140
.321
163/495
.329
312/104
6
.298
a.
b.
Explain how this demonstrates Simpson’s Paradox
Why does Simpson’s Paradox occur in this situation?
Statistics - Calc
A country exports crayfish to overseas markets. The buyers are prepared to
pay high prices when the crayfish arrive still alive. In order to help them plan
appropriately they have been keeping track of the number of deaths in each
shipment and have found how likely 1, 2, 3, 4, or 5 crayfish (per dozen) are to
die, as seen below.
x
a.
b.
c.
d.
0
1
2
4
5
>5
P(X = x)
0.54
0.36
0.15
k
0.01
0.00
Unfortunately, they can’t read the handwriting of their former intern who
made the table, so find k for them.
Over the long haul how many crayfish per dozen should they anticipate
losing?
Find the standard deviation and variance of the data
Suppose a rival tampered with their data and changed the # of crayfish
per dozen by using the algorithm 3x-2. How would this alter the mean and
variance? What about the standard deviation?
Statistics – No Calc
Statistics – No Calc
Consider the data set {8, 11, 12, 9, a}
a. Find the mean of the data set in terms of a
b. Give that the variance of the data set is 6
find the possible values of a
Statistics – No Calc
Suppose a, b, c, d, and e have a mean of 8.
Find the mean of 10 – a, 10 – b, 20 – c, 20 –
d, and 50 – e.
Statistics – No Calc
The data set {12, 13, 8, 10, 14, 7, a, b} has a
mean of 10 and a variance of 8.5. Find a and
b given a<b.