Transcript EXPECTED VALUE

Find Expected Value
Lesson 6.5
Page 355
EXPECTED VALUE
• Suppose you and a friend are playing a game. You
flip a coin.
• If the coin lands heads up, then your friend scores 1
point and you lose 1 point.
• If the coin lands tails up, then you score 1 point and
your friend loses 1 point.
• After playing the game many times, would you
expect to have more, fewer, or the same number of
points as when you started?
• The answer is that you should expect to end up
with about the same number of points. In other
words, you can expect to lose a point about 50% of
the time and win a point about 50% of the time.
• The expected value for this game is 0.
• A collection of outcomes is partitioned into n
events, no two of which have any outcomes in
common. The probabilities of n events occurring
are p1 , p2 , p3 …,pn ,where p1 + p2 + p3 …+ pn = 1.
• The values of the n events are x1 , x2 , x3, …, xn .
• The expected value E of the collection of outcomes
is the sum of the products of the events’
probabilities and their values.
• E = p1x1 + p2x2 + p3x3 + …+ pnxn
• You and a friend each flip a coin. If both coins land
heads up, then your friend scores 3 points and you
lose 3 points.
• If one or both of the coins lands tails up, then you
score 1 point and your friend loses 1 point.
• What is the expected value of the game from your
point of view?
Solution
• When the 2 coins are tossed, four outcomes are possible:
HH, HT, TH, and TT.
• Let event A be HH and event B be HT, TH, and TT.
• Note: all possible outcomes are listed, but NO outcome is
listed twice.
• The probabilities of the events are:
• P(A) = ¼
P(B) = 3/4
Solution Continued
• From your point of view the values of the events
are:
– Value of event A= -3
– Value of Event B=1
• Therefore, the expected value of the game is:
E = 1/4 (-3) + ¾ (1) = 0
• What is a “fair game”? A fair game is one where
each player has an equal chance of winning, and
the expected value is 0.
• You participate in a game show in which you
respond to questions that have 3 possible answers.
You gain $10 for each correct answer and lose $6 for
each incorrect answer. Every question must be
answered.
• If you do not know the answer to one of the
questions, is it to your advantage to guess the
answer?
Solution
• Step 1—Find the probability of each outcome.
Because each question has 1 right answer and 2
wrong answers, the probability of guessing correctly
is ___. The probability of guessing incorrectly is ___.
• Step 2—Find the expected value of guessing an
answer. Multiply the money gained or lost by the
corresponding probability, then find the sum of
these products.
• Expected value =
_____ · (1/3) + ______ · (2/3) = _______
• Is it to your advantage to guess an answer? Why or
why not?
Practice Problem
• A movie theater is giving away a $100 prize and a $50
prize.
• To enter the drawing, you need to buy a movie ticket
for $6. The ticket collectors will take the tickets from
the first 1000 guests, and after the movie ends, they
will randomly choose one ticket. If the number
chosen matches the number on your ticket stub, you
win 1st or 2nd prize. What is the expected value of
your gain?
• Step 1: Find the gain for each prize by subtracting
the cost of the ticket from the prize money.
• Step 2: Find the probability of each outcome.
There are 1000 tickets sold and the probability of winning
one of the prizes is ________. Because there are two
prizes, there are 2 winning tickets and _____ losing
tickets. So, the probability you will NOT win a prize is
__________.
• Step 3—Summarize the information:
– Gain, x
_____
_____ _____
– Probability 1/1000 1/1000 998/1000
• Step 4—Find the expected value by finding the sum
of each outcome multiplied by its corresponding
probability.
– V = __ (1/1000) + __(1/1000) + __(998/1000) = __
– The expected value of your gain is ___. This means that
you can expect to ___ an average of ______ for each
ticket you buy.
Homework Assignment:
Page 357 # 1, 2, 3, and 9