Probability
3.9 Expected Values
(ii) Your college has 75% women students. A student is selected at random and found to be color-blind. What is the probability that the student was a man?
(Assume the percentages for color-blindness are the same as in the general population.)
9. A test for a disease gives a positive reading inp% of cases when the disease is present, and also gives a (false) positive reading inp% of cases when the disease is absent. By coincidence, exactlyp% of the population suffer from the disease.
Prove that if a person tests positive, the probability that she has the disease is 50%.
150 3 Probability
100·0+300·1+300·2+100·3 100+300+300+100
= 300+600+300 800
= 1200 800 =1.5.
(The procedure is to find the mean, acting as though the expected frequencies were the actual frequencies that occurred.) We shall call this expected average the expected value of the experiment. The expected value might be called the theoretical mean of the experiment. We shall use μto denote a theoretical mean; if more than one variable is being discussed, the mean of a variableXwill be denotedμX.
In general, suppose an experiment has possible outcomesE1, E2, . . . , Ek, where Ei has probabilityP (Ei)=piand associated valuexi. If the experiment is repeated ntimes, the expected frequency ofEi isnpi, and the expected value of the experi- ment is
μ= k
i=1npixi k
i=1npi = k
i=1pixi k
i=1pi
= k i=1
pixi
since
k i=1
pi =1
.
Sample Problem 3.35. In an experiment, three balls are drawn at random from a sack containing four red and six blue balls. What is the expected number of red balls drawn?
Solution. WriteEi for the event thatired balls are drawn and define the value ofEi to bei. What is required is the expected value of this experiment. IfS is the sample space, then
|S| =C(10,3)=120.
If a selection containsired balls, it must contain 3−iblue balls. There areC(4, i) ways of selectingired balls from the four, andC(6,3−i)ways of selecting 3−i blue balls, so
|Ei| =C(4, i)C(6,3−i).
Therefore,
|E0| =C(4,0)· C(6,3)=20,
|E1| =C(4,1)· C(6,2)=60,
|E2| =C(4,2)· C(6,1)=36,
|E3| =C(4,3)· C(6,0)=4,
so
p0= 20
120, p1= 60
120, p2= 36
120, p3= 4 120 and
μ= 20·0+60·1+36·2+4·3 120
= 144 120 =1.2.
So the expected number is 1.2.
Your Turn. A committee of three is selected from a group of five girls and three boys. If the choice is random, what is the expected number of boys?
Random Variables
When an experiment has a value associated with each outcome, it is convenient to associate a variablexwith the experiment. If, for example, the value associated with outcomeE1isx1, then we say thatx =x1whenE1occurs; at any time,xequals the value associated with the current outcome. Thenx is called the random variable associated with the experiment. The expected value of an experiment with associated random variablexis also called the expected value or mean ofx.
In many gambling games, the player pays a fixed amount to play. The play can often be considered as an experiment, with the associated random variable being the payoff. The player’s expected net loss or gain is found by subtracting the cost of a play from the expected value of the variable. A fair game is one in which the expected net gain equals the price per play.
Sample Problem 3.36. A gambler flips an unbiased coin. If it shows tails, she loses. If it is a head, she flips again; if it falls tails she stops, but if it falls heads a second time, she flips once more. For three heads she wins $20, two heads wins
$6, and one head wins $2. How much does she expect to win? If she has to pay
$4 each time she plays this game, should she expect to win or lose in the long run? How much should be charged if this is to be a fair game?
Solution. LetEi denote the event thatiheads are thrown, andxi the payout for iheads. Half the time we expect the first flip to fall tails, soE0has probability 0.5. In the remaining cases, we expect to see tails at the second toss in 50% of cases, soP (E1)=0.5·0.5=0.25. And so on. We have
E0,probability 0.5,valuex0=0, E1,probability 0.25,valuex1=2, E2,probability 0.125,valuex2=6, E3,probability 0.125,valuex3=20.
152 3 Probability
So the expected value of the payout is
0.5·0+0.25·2+0.125·6+0.125·20=3.75.
So she expects to win $3.75 per play. If she pays $4 each time, she will have a net loss of 0.25 cents per play, so she expects to lose in the long run. If the game is to be fair, she should be charged $3.75 per play.
Your Turn. Two gamblers, A and B, each roll a die. If the total is 3 or 9, A pays B $5; if it is even, B pays A $2; otherwise there is no payoff. Who expects to win, and by how much?
Binomial Variables
Bernoulli trials and binomial experiments were discussed in Section 3.3. Briefly, a Bernoulli trial is an experiment in which there are exactly two possible outcomes, success and failure, and a binomial experiment is a sequence of Bernoulli trials where the probability of success is the same in each trial. The variable associated with a binomial experiment is the number of successes achieved. We writepfor the proba- bility of success in a single trial andq=1−pfor the probability of failure.
Theorem 8. Suppose a binomial experiment consists ofntrials, and the probability of success at each trial isp. Then the mean of the experiment isnp.
The proof appears later in this section.
Sample Problem 3.37. A racecar driver wins on average one out of every seven races. If he competes 35 times in the season, how many times does he expect to win?
Solution. With no information to the contrary, we assume that success or failure in one race does not affect success or failure in the next. So we can model the driver’s performance as a binomial experiment, in which each race is a Bernoulli trial and success equates with winning. In this model we haven =35,p = 17, andμ=35·17 =5.
Your Turn. A bowler averages one strike every five frames. If she bowls 40 frames, what is her expected number of strikes?
The Binomial Mean
This proof can be omitted at a first reading.
We saw in Section3.3that the probability of exactlyksuccesses in a binomial experiment is
pk =C(n, k)pk(1−p)n−k. So the mean is
μ= n k=1
kpk= n k=1
kC(n, k)pk(1−p)n−k. (The sum actually starts fromk=0, but the term withk=0 is zero.)
kC(n, k)pk(1−p)n−k= n!k
k!(n−k)!pk(1−p)n−k
= n!
(k−1)!(n−k)!pk(1−p)n−k
=n (n−1)!
(k−1)!(n−k)!pk(1−p)n−k
=npC(n−1, k−1)pk−1(1−p)n−k. So
μ= n k=1
npC(n−1, k−1)pk−1(1−p)n−k.
Suppose we defineh=k−1. The equation becomes μ=np
n−1
h=0
C(n−1, h)ph(1−p)(n−1)−h. From the binomial theorem,
n−1
h=0
C(n−1, h)ph(1−p)(n−1)−h=
(1−p)+p n−1, which equals 1 because(1−p)+p=1. So
μ=np.
Exercises 3.9 A
1. A fair coin is tossed three times and the number of heads is recorded. Supposex is the number of heads.
(i) CalculateP (x=0),P (x=1),P (x=2), andP (x=3).
(ii) What is the expected value ofx?
154 3 Probability
2. A game has four possible outcomesE1,E2,E3,E4. If outcomeEi occurs then the player wins $i. The probabilities of the outcomes areP (E1)= 25,P (E2)= P (E3)=P (E4)= 15. What does the player expect to win per play?
3. You play a game in which you toss two fair coins. If both land heads, you win
$10; if there is one head, you win $5; if both are tails, you win nothing. How much should you pay to play, if this is to be a fair game?
4. A player tosses a fair coin 10 times. Supposex is the number of heads. If x is even, the player wins $10; if it is odd, she loses $10. What is the player’s expected winnings?
5. An investor wishes to invest $100000 in a new dotcom company. She estimates that in the first year there is a 25% chance of making 100% profit, a 40% chance of making 50% profit, and a 35% chance of losing 80% of the investment. What is her expected gain (or loss) after one year?
6. A fair die is rolled 30 times. What is the expected number of sixes?
7. A test contains 10 multiple-choice questions, each with four answers. A student answers the test by random guessing. What is the expected number correct?
8. Two dice are rolled and the total is noted. Call the roll a “success” if the total is 9.
(i) What is the probability of a success?
(ii) The experiment is performed twice. Calculate the probability of x suc- cesses, forx=0,1,2.
(iii) What is the expected value ofx?
9. In a raffle with 100000 tickets the first prize is $20000, there are 10 second prizes of $1000, and 50 consolation prizes of $200. What is the expected value of a ticket?
10. A gambler bets $6 and rolls a fair die. If a six shows, he wins $9 (in addition to getting his original investment back). If a five or four shows, his money is returned with an additional $3. Otherwise he loses his bet. How much does he expect to win or lose per game?
Exercises 3.9 B
1. Among one strain of laboratory rats, 25% are born with weak hind legs. Letx denote the number with this defect in a random sample of three rats.
(i) Calculate the probabilitiesP (x=0),P (x=1),P (x=2), andP (x=3).
(ii) What is the expected value ofx?
2. A game has four possible outcomes E1, E2, E3, E4. If outcome Ei occurs then the player wins $i. The probabilities of the outcomes areP (E1) = 12, P (E2)= 14,P (E3)=P (E4)= 18. What does the player expect to win per play?
3. In a raffle with 1000 tickets the first prize is $200, there are five second prizes of
$50, and 10 consolation prizes of $5. What is the expected value of a ticket?
4. A gambler bets $2 and tosses three fair coins. If all three land heads, she wins $6 (in addition to getting her original bet back). If there are two heads, her money is returned. Otherwise she loses her $2. How much does she expect to win or lose per game?
5. A fair coin is tossed 240 times. What is the expected number of heads?
6. A jar contains three red and seven green marbles. Three marbles are drawn at random, without replacement.Eiis the event thatired marbles are selected.
(i) CalculateP (E1),P (E2), andP (E3).
(ii) What is the expected number of red marbles?
7. 1% of video cassettes are faulty. In a batch of 600, how many are expected to be faulty?
8. A state lottery sells 100000 tickets at $1 each. The first prize is $30000. There are four prizes of $4000, and 100 prizes of $100. What is the expected value of a ticket?
9. Most American roulette wheels have 38 numbered spaces: 1 through 36, 0, and 00. A number is chosen at random when the ball falls into that slot. Payoffs for bets are determined as though there were only 36 spaces—for example, if a player bets $10 that a specific number will be selected, say 15, and 15 is the winner, she receives $360 (including her initial bet); otherwise she loses her investment.
(i) What is the expected value of a $10 bet on a specific number?
(ii) Wheels at Monte Carlo usually have 37 slots, with no 00. The return is the same as in America. What is the expected value in this case?
10. On an American roulette wheel, as described in the previous question, the 0 and 00 spaces are colored green; half the others are colored red, and the remainder are black. It is possible to bet “red”; if a number in a red space is rolled, you receive your initial bet plus an equal amount. What is the expected value of a
$10 bet?
11. A drug manufacturer finds that 5% of its aspirin bottles contain more than the promised 100 tablets. In a batch of 10000 bottles, what is the expected number of those “overfull” bottles?
12. A box contains eight electrical switches, of which three are faulty. A switch is chosen at random and tested. If it is faulty, another is chosen and tested. The process continues until a nondefective switch is found. find the expected number of switches tested.
13. You play a game in which you toss three fair coins. If all three land heads, you win $18; if there is one head, you lose $6; if all three are tails, you lose $24.
There is no payoff for two heads.
156 3 Probability
(i) What is the expected value of this game?
(ii) Is the game fair?
(iii) If your answer to part (ii) is no, how much should you win or lose if two heads occur, in order to make the game fair?
14. Suppose a family has six children. Assuming boys and girls are equally likely, what is the expected number of girls in the family? What is the probability that this number occurs?
15. A coin is weighted so that heads are twice as likely as tails. It is tossed until either a tail occurs or three heads have occurred. What is the expected number of tosses?