Unit 4 Starters

1
Unit 4 Starters
2
Starter 7.1.1

• Suppose a fair coin is tossed 4 times. Find the
  probability that heads comes up exactly two times.

3
Answer

• A tree diagram will show that there are 16
  possible outcomes.
• The tree diagram also shows that 6 branches
  contain exactly two heads.
• Therefore P(two H) 6/16
• Another approach
• There are 2x2x2x216 outcomes from 4 flips
• Choose any two flips to be heads 4C26
• So P(two H) 6/16

4
Starter 7.1.2

• A 9-sided die has three faces that show 1, two
  faces that show 2, and one face each showing 3,
  4, 5, 6.
• Let X be the number that shows face-up.
• Draw the PDF histogram of X.

5
Answer

• What is the total area under the histogram?
• (3/9) (2/9) 4 x (1/9) 9/9 1

6
Starter 7.1.3

• What is the difference between a discrete random
  variable and a continuous random variable?
• There are two important ideas here
• Assume a certain continuous random variable X is
  known to be N(55, 1.2). If you randomly choose
  one observation of X, what is the probability
  that
• X lt 55
• X lt 53
• X gt 53
• Hint normalcdf(LB, UB, mean, s.d.)

7
Answer

• A discrete random variable has a finite number of
  outcomes that can only take on specific (usually
  integer) values. It can be represented by a
  table or histogram.
• A continuous random variable has an infinite
  number of outcomes. We can never list them all,
  and they typically include rational and
  irrational decimals. It can be represented by a
  density curve (only).
• P(Xlt55) .5 by definition 55 is the median
• P(Xlt53) normalcdf(0, 53, 55, 1.2) .0478
• P(Xgt53) 1 .0478 .9521

8
Starter 7.2.1

• An unfair six-sided die comes up 1 on half of its
  rolls. The other half of its rolls are evenly
  spread through the other 5 outcomes.
• If X is the number that comes up, write PDF for
  X.

9
Answer
X 1 2 3 4 5 6
P(X) .5 .1 .1 .1 .1 .1
10
Starter 7.2.2

• Calculate from formula the variance of these
  data
• 1, 2, 3, 4, 5

11
Answer
12
Starter 7.2.3

• Suppose I measure the heights of a class of
  fourth-graders and find the distribution of
  heights to be N(100, 2 cm).
• Then I have them all stand on top of a 10 cm high
  step and I measure again.
• What change would you expect in the mean?
• What change would you expect in the standard
  deviation?

13
Answer

• Since we added 10 to all values, the mean should
  increase by 10 cm.
• Adding 10 to all values did not change the
  spread it just changed the location if we
  plotted on an axis. So we expect the standard
  deviation to remain unchanged at 2.

14
Starter 7.2.4

• A bag contains 4 red marbles and one white
  marble. Two marbles are chosen without
  replacement. What is the probability that they
  are both red?
• A worker is paid each week in the following
  manner He draws two bills from a bag which
  contains four 20 bills and one 100 bill. What
  is the expected outcome of his average weekly pay
  over the long run?

15
Answer

• Let X the amount of weekly pay
• Notice that X can only be 40 or 120
• There are 5C2 ( 10) ways to draw two bills from
  the five in the bag
• There are 4C2 ( 6) ways to draw two 20 bills
  from the four in the bag
• So P(40) 6/10
• There are 4 ways to draw one 100 and one 20
• So P(120) 4/10
• E(X) 40 (6/10) 120 (4/10) 72

16
Starter 8.1.1

• Heres a game you will like Lets bet a dollar
  on this proposition I will roll a fair die
  once. If it comes up 1 or 2, I win. If it comes
  up 3, 4, 5, or 6, you win!
• What is the probability that I win?
• What is the probability that I lose?
• How much should we each bet to make this a fair
  game?

17
Answer

• P(win) 2/6 or 1/3
• P(lose) 4/6 or 2/3
• Odds that I win are 24 (or 12) against, so you
  should bet 2 and I bet 1
• Note fair in this context means E(x) 0,
  where x represents my net winnings.
• E(x) (2)(1/3) (-1)(2/3) 0, so its fair

18
Starter 8.1.2

• Five white marbles are on a table. Two of them
  are to be painted with a W and the rest will be
  painted with a L.
• How many ways are there to choose the two marbles
  to be painted W?

19
Starter 8.1.3

• A baseball player bats .325 over a full season.
  If he bats 5 times today, find the probability he
  gets at least 3 hits.

20
Answer

• This is a binomial setting, so use the binomial
  CDF of at most 2 hits, then subtract from 1
• 1 binomcdf(5, .325, 2) .197

21
Starter 8.1.4

• A manufacturer produces a large number of
  toasters. From past experience, he knows that
  about 2 are defective. In a quality control
  procedure, we randomly select 20 toasters for
  testing. We want to determine the probability
  that no more than one of these toasters is
  defective.
• Is this a binomial setting? Justify your answer.
• Find the probability that exactly one toaster is
  defective.
• Find the probability that at most one toaster is
  defective.
• Find the mean and standard deviation for the
  problem.

22
Answer

• Two outcomes (good / defective), a fixed number
  of trials (20), independent trials, probability
  is fixed (2), so it is binomial.
• P(X1) binompdf(20, .02, 1) 27
• P(X?1) binomcdf(20, .02, 1) 94
• µ np (20)(.02) .4
• So we expect on average .4 defectives in each
  group of 20
• s vnpq v(20)(.02)(.98) .626

23
Starter 8.2.1

• Fred Funk hits his tee shots straight most of the
  time. In fact, last year he put 78 of his tee
  shots in the fairway.
• In yesterdays round, he hit only 7 of 14 shots
  in the fairway.
• What is the likelihood that he would have that
  bad a day (or worse) from the tee?

24
Answer

• This is a binomial setting
• There are 14 trials where he succeeds or fails at
  hitting the fairway
• All trials are independent with fixed probability
• To do this poorly (or worse) means he hits at
  most 7 fairways.
• He could hit 7 or 6 or 5 or
• Binomcdf(14, .78, 7) .02
• So there is only a 2 chance he will do that
  poorly.

25
Starter 8.2.2

• The SAT Math and Verbal sections are both
  designed to be approximately normal with a mean
  of 500 and standard deviation of 100.
• If we defined a new measure (TOTAL) by adding the
  scores on the two sections, what would you expect
  the mean and standard deviation of TOTAL to be?
• (Assume math and verbal are independent)

26
Answer
27
Starter

• Write the characteristics of the binomial
  setting.
• What is the difference between the binomial
  setting and the geometric setting?

28
Answer

• The binomial setting has 4 characteristics
• There are only two possible outcomes
• Each trial has fixed probability of success
• Each trial is independent of all other trials
• There are a fixed number of trials the variable
  of interest is the number of successes
• The geometric setting has the same first three
  characteristics. The difference is in the
  fourth there are an unknown number of trials
  trials stop after the first success the variable
  of interest is the number of trials until the
  first success