Chapter 6 - Probability and Simulation: The Study of Randomness
Name
6.1 Homework: Give answers as the EXACT fraction or decimal. If you must round off,
round to three significant digits.
Coin Toss
1. Hold a penny upright on its edge under your forefinger on a hard LEVEL surface, then snap it
with your other forefinger so that it spins for some time before falling. Repeat this experiment
50 times.
a. What is the probability of heads?
b. What is the year of your penny?
c. Do different pennies show the same results? Compare your results with the rest of the class.
2. A very successful football coach once explained why he preferred running the ball to passing
it: "When you pass, three things can happen [completion, incompletion, or interception] and two
of them are bad." Can we infer that there is a 2/3 probability that something bad will happen
when a football team passes the ball?
No. When the quarterback passes the ball, he probably completes more the 1/3 of his passes.
Each of the events is not equally likely to occur.
3. The most important skating event in the Netherlands is the Elfstedentocht, a race over 124
miles of canals through 11 Dutch cities. This race is only held if the entire course is covered by
ice at least 8 inches thick. During the 96 years from 1900 to 1995, the race was held 14
times. Based on these data, what is your estimate of the probability that the race will be held
next year?
7/48
4. You are playing draw poker and are dealt four spades and a heart. (Fifty-two cards in a deck,
13 cards of each type.) If you discard the heart and draw a new card, what is the probability that
this new card will be a spade, giving you a flush? (Assume that there are no other players, since
it can be shown that your chances do not depend on whether there are other players, as long as
you do not know what cards they have been dealt.)
9/47
5. a. If a person is randomly selected, find the probability that his or her birthday is January 1st,
ignoring leap years.
1/365
b. If a person is randomly selected, find the probability that his or her birthday is in
November. Ignore leap years.
6/73
6. In a study of blood donors, 225 were classified as group O and 275 had a classification other
than group O. What is the approximate probability that a person will have group O blood?
9/11
7. Which of the following values cannot be probabilities?
0,
0.0001,
-0.2,
3/2,
2/3,
square root of 2,
square root of 0.2
8. A Gallup survey found that 228 people brushed their teeth once a day, 672 people brushed
twice a day, and 240 people brushed three times a day. If one of the respondents is randomly
selected, find the probability of getting someone who brushes their teeth three times a day.
4/19
What is the probability of selecting someone who brushes two or more times a day? 4/5
9. Establish a number correspondence that aids a simulation of a 75% chance for the following
situations:
a. a coin Flip the coin 2 times. You will have the following sample space: {HH, HT, TH, TT} If
you flip HH, HT, or TH, these will correspond to the 75%.
b. a six-sided die Only use the numbers 1-4, 1-3 will correspond to the 75%. If you roll a 5 or a
6, ignore them.
c. a random digit table Assign numbers 00-74 for the 75%, 75-99 for the remaining numbers.
d. a deck of cards Assign 3 of the suites, such as spades hearts and clubs, as the 75%. The
remaining suite will be the other 25%.
10. Brickel, a Juab basketball player made 70% of her free throws in a long season last year. In
a tournament she shoots 5 free throws late in the game and misses three of them. The wasp fans
think she was nervous, but the misses may simply be chance. Use a simulation to determine if
this is unusual.
a. Describe how to simulate a single shot if the probability of make each shot is 0.7. Then
describe how to simulate 5 independent shots.
Step 1: State the problem – Was it unusual for Brickel to miss 3 out of 5 free throws?
Step 2: State the assumptions – We must assume that each free throw shot is independent from
the other shots. Each shot taken is also equally likely to occur.
Step 3: I used the random number generator on the calculator. RandInt(1,10,5). The numbers 17 represent a made basket. The numbers 8-10 represent a missed basket.
b. Simulate 50 repetitions of the 50 shots and record the number of misses on each repetition.
Use table B, starting at line 125. What is the approximate likelihood that Brickel will miss 3 or
more shots?
Step 4: Simulate many repetitions
# of missed shots/trial: 2, 3, 3, 2, 0, 0, 0, 0, 2, 0, 1, 2, 1, 2, 0, 1, 1, 0, 2, 2, 2, 1, 2, 1, 3, 0, 2, 1, 1,
1, 0, 1, 2, 1, 2, 3, 2, 1, 2, 2, 1, 3, 2, 2, 2, 1, 3, 1, 2, 3
I performed 50 trials. Of the 50 trials, 7 missed three or more shots. So, we would expect
Brickel to miss three shots out of 5 about 14% of the time.