Problem of the Day
Consider a dice game: no points for rolling a 1, 2, or 3; 5 points for a 4 or 5; 50 points for a
6. Create a probability model. Find the expected value and the standard deviation. Interpret the standard deviation.
Problem of the Day
Your company bids for 2 contracts on dry wall. You believe the probability of getting #1 is .8. If you get contract #1, the probability that you get #2 is .2. If you do not get #1, the probability of getting #2 is .3. Are the contracts independent? Find the probability that you get both. Find the probability that you get neither.
Problem of the Day
Why does an insurance company ever consider giving individuals life­insurance that could cost them $500,000 if the person dies?
Chapter 16 Random Variables
Greedy Pig
Everyone needs to stand. I will roll a die. Everyone standing gets the points rolled. You can sit down and keep your points(you stop accumulating) whenever you choose. If I roll a 5, everyone still standing gets 0 and the game is over. The person(s) with the most points wins.
Chapter 16 Random Variables
Random Variable: numeric value assigned to an outcome of a random event
Discrete random variable:
Continuous random variable:
Probability model: collection of all possible outcomes and their probabilities
Policy Holders State
Dead
Disabled
Neither
Payout(x)
$10,000
$5,000
$0
Probability P(X)
Expected Value(mean/center of a random variable)
It is important that you have a valid probability model to begin with.
Standard Deviation(recall)
­takes into account how far each value is from the mean(center)
­for symmetric data
­need to calculate the variance of the data then use it to find the standard deviation
Find the expected value of a single random payout.
Policy Probability Holders Payout(x)
x p(x)
p(x)
State
Dead
$10,000 1/10000
Disabled $5,000 2/10000
Neither
$0
9997/10000
Standard Deviation for a Discrete Random Variable Policy Probability Holders Payout(x)
P(X)
State
Dead
$10,000 1/10000
Disabled $5,000 2/10000
Neither
$0
9997/10000
What the heck does that mean? For real Mr. Stalter, that makes zero(maybe less) sense to me!!!!
Consider the idea that you pay $120 a year for this life insurance policy. What is the insurance company "making or losing" each year if they
insure __________ people?
Valentines dinner at the Quiet Nook comes with this possibility of getting a discount. With the bill, you receive the opportunity to draw 1 of the 4 aces that the waiter brings. If you draw the Ace of hearts you get $20 off. If you draw the Ace of diamonds you get to choose another card and if you choose the Ace of hearts you get $10 off
If you draw a black Ace you pay the full bill
Construct a probability model for the situation and find the expected value.
Notation(parameters)
2
σ
σ
You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6 the second time, you win $50. If not, you lose. a) Create a probability model for the amount you win. b) Find the expected amount you’ll win. c) What would you be willing to pay to play this game? A couple plans to have children until they get a girl, but they agree that they will not have more than three children even if all are boys. (Assume boys and girls are equally likely.) a) Create a probability model for the number of children they might have. b) Find the expected number of children. c) Find the expected number of boys they’ll have. Tree diagrams
A small software company bids on two contracts. It anticipates a profit of $50,000 if it gets the larger contract and a profit of $20,000 on the smaller contract. The company estimates there’s a 30% chance it will get the larger contract and a 60% chance it will get the smaller contract. Assuming the contracts will be awarded independently, what’s the expected profit? Changing Means and Variance
What if the Quiet Nook had $5 off coupons in the local paper(1 per table)? What is the new mean and variance?
Changing Means and Variance
What if the Quiet Nook doubles the discounts the following year to increase the number of people with reservations? What is the new mean and variance? What about standard deviation?
Recap
When adding/subtracting means and variances by a constant
Mean
Variance
Standard Deviation
Recap
When adding means and variances
Mean
Variance
Standard Deviation
Recap
When multiplying means and variances by a constant
Mean
Variance(multiply, but wait)
Standard Deviation
The American Veterinary Associaon claims that the annual cost of medical care for dogs averages $100 with a standard deviaon of $30, and for cats averages $120 with a standard deviaon of $35.
a. Find the expected value for the annual cost of medical care for a person who has one dog and one cat.
b. Find the standard deviaon for the annual cost of medical care for a person who has one dog and one cat.
c. Suppose that a couple owns four dogs.
i. Find the expected value for the annual cost of medical care for the couple’s dogs.
ii. Find the standard deviaon for the annual cost of medical care for the couple’s dogs.
Practice Problem
Suppose a used car dealer runs autos through a two­stage process to get them ready to sell. The mechanical checkup costs $50 per hour and takes an average of 90 minutes, with a standard deviation of 15 minutes. The appearance prep (wash, polish, etc.) costs $6 per hour and takes an average of 60 minutes, with a standard deviation of 5 minutes. What is the mean and standard deviation of the total "clean­up" time?
If both times are normally distributed, what is the probability that it will take longer to do the appearance prep than the mechanical checkup?
Russel's Cycle and Fitness receives their bikes in boxes. The means and standard deviations of the 3 different phases of assembly(in minutes) are as follows....
Phase
Unpacking Mean Standard Dev
3.5
.7
Assembly
21.8
2.4
Tuning
12.3
2.7
If all phases are normally distributed, find the following.......
What is the probability that unpacking and assembly will take more than 25 minutes?
What is the probability that their Superstar Employee Champ Bikesmasher can get a bike ready for sale in 30 minutes?
Homework: pg 383:1­3,9­11,15,19, 21­23,25,27,31,33,36,41
Readings and examples pgs 366­382
An insurance company sells life insurance of $15,000 for a premium of $310 per year. Actuarial tables show that the probability of death in the year following the perchase of this policy is .1%. What is the expected gain for this type of policy? What is the expected profit if the insurance company insures 100 people?