PROBABILITY
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PROBABILITY
Exercise P1
Suppose we each flip a coin,but your coin has "heads"on both sides.Such coins can be purchased at magic and novelty stores,I am told.Specify the possible outcomes,and compute the probability of each outcome.Write your answers below.
Exercise P2
Here's a simple game with two dice.I take the even numbers and you take the odd numbers.I win if an even number turns up,and you win with an odd number.(Note that there are six possible even numbers,but only five possible odd numbers.)
a. Does either one of us have an advantage?
b. Why or why not?
C. What did I mean by "advantage?"
Exercise P3
Iroll two dice,and the idea is to get"doubles"-to have the two dice match.I get two rolls.IfI don't get doubles on the first roll,then I can roll both dice again,or I can just roll one of the dice again in the hope of matching the other one.
a. What is the probability that I will roll doubles on the first roll?Study the dice table above.
b. IfI don't get doubles on the first roll,should I roll both dice again,or should I just roll one again?
C. What is the probability that I can win the game;i.e.,roll doubles given that I have two chances?
Is this a "fair"game?What would one mean by fair game?
Exercise P4
a. Look at my nice dice table above.What is the probability that you will roll a four or lower?
b. What is the probability that you will roll a 10 or higher?
C. What is the probability that you will roll a 7 on the first roll and a 6 on the second roll?(Remember the multiplicative rule here.)
Kxercise P5
You roll one die only(the usual type of die,with the number 1 through 6 on its sides).What is the mean for the outcomes?
Exercise P6
a. You are a farmer, and you worry about the weather because your income depends upon it.We assume that we know the probability table.Here is the probability table that applies:
Weather Probability Income
normal .8 $100,000
drought .19 $0
tornado .01 -$100,000
What is the mean value for your income?As noted above,this would also be called your expected income.
b You are a school teacher.The probability table that applies is:
Weather norma |
Probability .8 |
Income $50,000 |
drought |
.19 |
$50,000 |
tornado |
.01 |
$50,000 |
What is your "expected"income?
Exercise P7
I have another funny die with the following numbers on the six sides:0,2,3,3,4,
6.(Contrast this to my other die with numbers on the sides of1,2,3,3,4,5.My new die has greater dispersion of the outcomes.)
a. Compute the mean of the outcomes for this new die.Write your answer below.
b. Compute the variance of the outcomes.
C. Compare these results to the mean and variance for the die with numbers on the side of 1,2,3,3,4,5.
Exercise P8
Compute the variance of the roll of two dice(the normal dice).I'll help you get started;
Variance =(2-7}(1/36)+(3-7)}(2/36)+ … …
(I know that this is a big computation,but I promise that it's the most complicated one that you'll do in this manual.)Show your computations below.
Exercise P9
a. Using the table for the normal distribution,what is the probability that a man chosen at random will have a height that is 6'2"or greater?
b. What is the probability for height 5'11"and greater?
C. What is the probability of height less than 5'8"?
d. What is the meaning of the third column in the table (labeled o)?
Exercise P10
Suppose we are told that the mean height for men in Iceland has increased to 5'10". The variance is still the same (9).Make up a new normal distribution table like the one I made for a mean height of 5'8".
Exercise P11
Now suppose we are told that the variance of heights for men in Iceland has declined to 4(standard deviation of2).Draw up a new normal distribution table like the one I did that used a standard deviation of3.The mean height is still 5'8".(Very Big Hint:Remember that normal distribution table is completely described by the mean and standard deviation,and that the probability of a man being within one standard deviation of the mean is always .6826.)
2024-01-12
Math and Statistics