Chapter 17
STA 200 Summer I 2011
Flipping Coins
• If you toss a coin repeatedly, you expect it to
come up heads half the time.
• Suppose you toss a coin 10 times.
– In this scenario, you “expect” to get 5 heads.
– Would it be that unusual to get 4 heads or 6 heads
instead? What about 3 heads or 7 heads?
Fundamentals
• Chance Behavior:
– unpredictable in the short run
– predictable pattern in the long run
• Random Phenomenon:
– individual outcomes are uncertain
– regular distribution of outcomes in a large number
of repetitions
Coin Tossing (cont.)
• When repeatedly tossing a coin, the
proportion of heads will vary quite a bit at
first.
• After more and more tosses, the proportion
will get close to 0.5 and stay there.
• Probability is an extrapolation of what would
happen in an infinitely long series of trials.
Probability
• The probability of any outcome of a random
phenomenon is a number between 0 and 1 that
describes the proportion of how often the outcome
would occur in a long (infinite) series of repetitions.
• If an outcome never occurs, it has probability 0.
• If an outcome always occurs, it has probability 1.
• Any other type of outcome has a probability
somewhere between 0 and 1.
Myth #1 – Short-Run Regularity
• The idea of probability is that a random event (like flipping a coin) is
regular in the long run. In the short run, this is not the case.
• For example, if a coin is tossed 10 times, the outcome HTTHTHTTHT
looks more probable than HHHHTTTTTT. However, both are equally
probable: both outcomes are 4 tails in 10 flips. Even though heads
and tails are equally probable, they don’t have to come close to
alternating in the short run.
• In other words, the coin has no memory. The probability of the
coin coming up heads on any given flip is 0.5, regardless of what has
already happened.
Myth #2 – Law of Averages
• Usually, when the phrase “law of averages” is
used, there is an implication that one or more
outcomes have somehow become more likely.
• For example: If a coin is flipped six times, and
comes up heads each time, then by the “law of
averages” it’s more likely for the coin to come up
tails on the seventh flip, in order to compensate
for the imbalance between heads and tails.
Law of Averages (cont.)
• The six consecutive outcomes of heads will never be
compensated for; instead, they will be overwhelmed in the
long run.
• If the coin is flipped 1,000 times, the results of the first 6
flips will be overwhelmed by the results of the following
994.
• The law of averages doesn’t work for the same reason that
short-run regularity doesn’t work: the individual coin flips
are independent of each other, so the probability of a coin
coming up heads on any given flip is unaffected by what
has previously occurred.
Law of Large Numbers
• There is something called the “law of large
numbers,” which states that the proportion of a
specific outcome (like a coin coming up heads)
will tend to stabilize as the number of trials
increases.
• However, this isn’t the same thing as saying that,
for example, the number of heads will tend to get
closer to the number of tosses. This may or may
not occur.
Law of Large Numbers (cont.)
• For example, the following results are certainly
possible:
difference
between # of
heads and
half the # of
tosses
# of tosses
# of heads
proportion of
heads
10
4
.4
1
100
47
.47
3
1,000
511
.511
11
10,000
5,074
.5074
74
Personal/Subjective Probability
• A personal probability is determined by a person making a
judgment regarding the likelihood of an event.
• The events of interest for personal probability are usually
ones that are difficult to repeat (or not repeatable at all).
• The value of a personal probability will vary from one
person to another.
• Personal probabilities can be found in weather forecasts,
stock price projections, and sports predictions.