Back to the Table of Contents
Statistical Probabilities and Distributions
Homework for Prob./Dist. Lesson 8
| \
| 1 |
2 |
3 |
4 |
5 |
6 |
|---|
| 1 | | | | | | |
|---|
| 2 | | | | | | |
|---|
| 3 | | | | | | |
|---|
| 4 | | | | | | |
|---|
| 5 | | | | | | |
|---|
| 6 | | | | | | |
|---|
- Consider again the random experiment consisting of rolling
two dice, a red and a green, as in lesson 1.
However, this time, instead of adding the pips, subtract the smaller
from the larger, forming the unsigned difference. Perhaps it would
be instructive to start by filling in the table at right.
Now generate a table tallying for each outcome (0 for doubles
up to 5 for a six and a one) how many of the 36 outcomes result
in that particular value. (One might note the triangular numbers involved,
since a square is the sum of two consecutive triangular numbers.)
From this please form the probability for each value,
then create a bar graph for the distribution.
Finally, note the mode or most probable outcome and
calculate the expected value by summing the product of the
value with its probability.
- Statistics show that about 8% of all human males exhibit some form of
color-blindness.
Suppose 20 males are selected at random.
Let x be the number who are color-blind,
and let P(x) be the probability that
x of them are color-blind.
Using the binomial distribution, TI-83+ program binomial,
or TI-83+ function binompdf(20,.08), calculate
P(0), P(1), P(2), P(3), and P(4).
Plot a graph of this probability distribution.
What is the probability that 5 or more males in a group of 20 are
color-blind.
How might color-blindness affect the military?
- Tom Bone plays a musical solo.
He is quite good and figures his probability of playing any one note
right is 99%. The solo has 50 notes.
What is his probability of:
- Getting every note right?
- Making exactly one mistake?
- Making exactly two mistakes?
- Making at least two mistakes?
- What probability per note is necessary for Tom to have a 95% probability
of getting all 50 notes correct?
(Logs or roots can be used to reduce an exponent.)
- Suppose a table group decides to do a little penny gambling.
Cards will be drawn at random from a standard 52-card deck.
For each card drawn you pay a quarter ($0.25).
The following payoffs apply: Ace (get $1.75), Face card (get $0.50),
Any other card (get nothing).
Calculate the expected value for this game.
Does this mean you would expect to gain or lose money, in the long run?
- Dudley Do Wright will get paid by his grandfather for
good grades in his college prep classes.
His grandfather, however, requires a contract, in advance.
Grandpa offers to pay $100 if Dudley makes all A's, or $10 per A.
Dudley estimates his probabilities of making A's as follows:
Algebra II: 0.75, Chemistry: 0.65, Computers: 0.86, English: 0.96.
- Calculate Dudley's expected value if he chooses $10 per A.
- Calculate Dudley's probability of making all A's.
- Calculate Dudley's expected value if he chooses $100 for all A's.
- Which offer should Dudley choose and why?
- Many tests have guessing penalties to correct for random guessing.
The Advanced Placement exams, for which our seniors prepare specifically
for the AP Calculus AB is a fine example. Other tests, such as SAT, ACT,
or the GRE may be similar.
- Each question is multiple choice with 5 choices.
If you guess randomly, what is the probability of getting a correct answer?
- Suppose you are awarded 1 point for each correct answer but lose
¼ for each wrong answer. What is your expected value for
any question you randomly guess on?
- Suppose you can eliminate one of the choices as clearly wrong.
Now what is your expected value if you randomly guess between the remaining
four?
- Repeat the prior subquestion, eliminating two of the five choices.
- Repeat the prior subquestion, eliminating three of the five choices.
- Is it really worth while guessing?