Warm up:

Solve each system (any method)

W-up 11/4• 1) Cars are being produced by two factories, factory 1

produces twice as many cars (better management) than factory 2 in a given time. Factory 1 is know to produce 2% defectives and factory 2 produces 1% defectives. A car is examined and found to be defective, what is the probability it was produced by factory 1?

• 2. evaluate b(7,4;.20)

• 3. A fair coin is tossed 8 times, what is the probability of obtaining at least 6 heads?

Answers: 1. 80% 2. 2.87% 3. 14.45%

8.3 EXPECTED VALUESWBAT compute expected values in addition to solving application problems involving expected value.

Consider a coin flipping game: If heads shows, you lose

$1. If tails shows, you win $2.• Let E be our expected value.

• Where ½ is the probability of getting Heads or Tails

*So you expect to win an average of $.50 on each play.

Expected Value:• S = Sample Space

• is assigned payoff .• The Expected Value E corresponding to the payoffs is:

Steps to compute E:• Partition “S” into the “A” events.• Determine the probability of each event (Sum of

probabilities should = 1).• Assign payoff values “m”.• Calculate.

Compute the expected value:Outcome

Probability 1/3 1/6 1/4 1/4

Payoff 1 0 4 -2

• SS: {}• Probability: Given• Payoff: Given

• = $.83

A player rolls a die and receives the # of $ = to the # of dots on the die. What is the expected value to play?Roll #1 #2 #3 #4 #5 #6

Probability 1/6 1/6 1/6 1/6 1/6 1/6

Payoff $1 $2 $3 $4 $5 $6

If E = 0 then the “game” is fair

A lab contains 10 microscopes, 2 are defective. If 4 are chosen what is the Expected value of Defective?

Probabilities of 0,1, or 2 defectives:

Assign payoffs of 0 (no defective)1,2 since we are determining the expected #:

• Talk about the answer. • Can’t have 4/5 of a microscope?• We can interpret this to mean that in the long run, we will

average “just under 1 defective microscope”

Expected Value of Bernoulli Trials:• With “n” trials the expected # of successes is:

E=np

*Where “p” is the probability of successes on any single trial.

MC Test contains 100 questions each w/ 4 choices. What is the expected # of correct guesses?• Answer: 25

• So using Bernoulli to explain:

HW WS: 8.3; #s 1-17odd,21, 25, 27