Download AP Statistics What is Expected Value and Why Should I Care

AP Statistics
What is Expected Value and Why Should I Care?
Games of chance involve taking a risk to win a prize. Whether or that risk is worth taking depends
on two things: 1.The value of the prize and 2. The probability of winning it
Expected Value is a way of measuring the combination of those two factors.
Consider the following game: We flip a coin. If it lands Heads, I pay you $1. If it lands Tails, you
pay me $5. Is this a fair game? That turns out to be an ambiguous question. On the one hand,
each of us has the same probability of winning. On the other hand, the rewards for winning are
not the same.
Enter Expected Value. To find it, just multiply each possible outcome by the probability that it will
1
1
occur. Your expected value =  $1   $5   $2 That means over time you stand to lose an
2
2
average of $2 for every time you play the game. Suddenly, fair doesn’t sound so fair, does it?
How about this game: When you play The Pennsylvania Daily Number, you choose a 3-digit
number (000 – 999) and you indicate how much money you’d like to wager ($.50 to $50). If your
number matches the number chosen – all three digits in the proper order – the payout is 500 to 1.
That is, you are rewarded $500 for each $1 you bet.
Question: Does it pay to play? Answer: Expected Value. Say you bet $1. If your number comes
up, you win $500. If not, you forfeit your dollar. Actually, you forfeit your dollar in both cases, so
1
999
your reward for winning is really $499. Your expected value =
 $499  
 $1  $.50
1000
1000
That means, over time, you stand to lose an average of $.50 for every time you play the lottery.
Suppose you play your “lucky” number every day. . . That means you stand to lose about $182 per
year! How lucky do you feel now?
With a little practice, you can expect that this will all become clear. Along the way, we’ll meet
some new friends. . .
Expected value can be thought of as the mean of a probability distribution:
n
E(X)   x i  Pr(X  x i )   X
i1
The variance (remember variance) of a probability distribution is a measure of its spread:
n
Var(X)   (x i   X )2  Pr(x i )  X2
i1
The standard deviation of a probability distribution is the square root of the variance:
SD(X)  Var(X)  X2  X
When adding or subtracting independent random variables:
E(X  Y)  E(X)  E(Y)
and
Var(X  Y)  Var(X)  Var(Y)
Note When we talk of probabilities, we are describing the longterm distribution of events. If we
flip a coin, we expect it to land Heads 50% of the time. That’s because the two outcomes, heads
and tails, are taken to be equally likely. Enough coins have been flipped over time to lead us to
assume they tend to land heads and tails in equal proportion. Things tend to even out in the long
run. This is sometimes referred to as the Law of Large Numbers.
Law of Large Numbers – A Cautionary Tale. Just because things tend to even out in the long run,
doesn’t mean that they will in your lifetime. If you flip a coin and get 10 heads in a row, don’t be
fooled into thinking that your luck is more likely to change on the next flip. While it is true that the
relative frequency of heads and tails tend toward 50-50 balance as the number of trials increases,
the next coin flip encounters no pressure whatsoever to compensate for any imbalance in the
relative frequency of heads and tails you may have experienced thus far. The probability of the
next flip landing heads is still 50%.