Law of Large Numbers

1
Law of Large Numbers

• For a repeated event A where the results of each
  trial is independent of the previous trialsAs
  the process is repeated through many trials, the
  proportion of the trials in which event A occurs
  will be close to the probability P(A).The
  larger the number of trials, the closer the
  proportion should be to P(A)

2
The Law of Averages
7-C
Below are the results of a computer simulation of
rolling a single die. The theoretical probability
of rolling a 1 is 1/6 0.167 (blue line) The
pink plot represents the actual proportion of 1s
from the simulation.
3
Expected Value

• The more times an independent trial is repeated,
  it trends towards its theoretical probability.
• If we flip a coin 100 times, we should expect 50
  heads and 50 tails
• If we roll a die 600 times, we should expect 100
  rolls of each number
• If we pull a card from the deck (with
  replacement) 4000 times, we should expect 1000 of
  each suit

4
Expected Value Formula

• Suppose we have 2 events, each with their own
  probability. The expected value is
• expected event1 event1 event2
  event2 value value prob. value
  prob.
• We take each event and multiply it with its
  probability and then sum up all of the products

5
Expected Value example

• Family with 3 children Expected number of
  girls
• P(0 girls) 1/8 P(1 girl) 3/8P(2 girls)
  3/8 P(3 girls) 1/8
• 0 girls P(0 girls)
  1 girl P(1 girl) 2 girls P(2 girls)
  3 girls P(3 girls)
• Expected 01/8 13/8 23/8
  31/8
• of girls (0 3 6 3)/8
• 12/8 1.5 girls

6
How do we use expected value?

• How can we expect to have 1.5 girls with a family
  of 3 children?
• If we look at 100 families of 3 children, how
  many girls would we expect there to be?Each
  family has, on average, 1.5 girls100 families
  1.5 girls each family 150 girls

7
Game application

• Someone picks a number from 1 to 10 at random and
  you have to try to guess it correctly. Correct
  guess win 50Incorrect guess lose
  10Should you play? What is the expected value
  of this game?

8
Insurance application

• Suppose an insurance policy pays 50,000 for a
  particular type of injury in the workplace.
  Using existing data, the empirical probability
  this type of injury occurs is 1 in 500. If the
  insurance company charges 150 for its policy,
  should they expect to make a profit?Expected
  value 1501 (-50,000)1/500 150 -
  100 50 expected profit / policy

9
Expected Value Exercises

• Find the expected value in each of the following
  cases
• Game roll a single die if a 1 or 2 comes up,
  you win 3. If a 3 or 4 comes up, you win 1,
  and if a 5 or 6 comes up, you lose 5.
• Insurance Insurance company charges 200/policy.
  Based on previous data, an average of 1 in 50
  policyholders file a 5,000 claim and 1 in 100
  policyholders file a 10,000 claim.
• Waiting times Based on previous data, a
  customer service call center has a wait of 3
  minutes 40 of the time, 10 minutes 50 of the
  time and 15 minutes 10 of the time.