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Elementary Statistics
Triola, Elementary Statistics 11/e
Unit 2 Probability Basics Continued
The Law of Large Numbers
Even though the probability of getting a heads on a flip of the coin is 0.5, you would not expect to get
exactly 5 heads were you to flip a coin ten times.
According to the law of large numbers, the more times you flip coin, the more times the number of
heads will approach one half of the total number of flips. If you were to flip a coin 10 times, you might
for example get 6 heads, the ration being 0.60. However, if you were to flip the same coin 1000 times,
you might get 540 heads, the ratio being 0.54. 0.54 is closer to the theoretical ration of 0.5 than 0.6 is.
The Complement
The complement of event 𝐴, denoted 𝐴̅, is the set of all events in the sample space other than the
event 𝐴 itself. For example, consider the experiment of flipping a coin twice. Let the event 𝐴 represent
getting two heads in a row, i.e. 𝐴 = {𝐻𝐻}.
Question #1
What is the set corresponding to 𝐴̅? Remember to include all the elements of the sample space other
than HH.
We say that 𝐴 and its complement, 𝐴̅, are mutually exclusive or disjoint, because as sets, they do not
contain any elements in common, i.e. there is no overlap between the two sets. Also, it is not possible
for two mutually exclusive events to occur at the same time. Later on we will be able to easily show the
following relationship between an event and its complement,
̅ ) = 𝟏. 𝟎 − 𝑷(𝑨)
𝑷(𝑨
Question #2
Let’s verify the formula, 𝑃(𝐴̅) = 1.0 − 𝑃(𝐴).
a. What is the probability of getting two heads in a row, P(HH)?
b. What is the probability of getting either a TH, HT, or TT which is the complement of HH?
c. Now, if we let 𝐴 = {𝐻𝐻} 𝑎𝑛𝑑 𝑃(𝐴̅) = {𝐻𝑇, 𝑇𝐻, 𝑇𝑇} verify that 𝑃(𝐴̅) = 1.0 − 𝑝(𝐴).
The Probability of “At Least One”
Let’s suppose that a family is planning to have three children, and they wish to know the probability of
getting at least one girl. The sample space of all possibilities looks like,
{GGG, GGB, GBG, GBB, BBB, BBG, BGB, BGG}
From this you can see that the probability of having at least one girl is 7 out of 8 or 7/8. Typically,
however it can be difficult to enumerate all the cases of at least one, but it is much easier to enumerate
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Unit 2
Probability Basics Continued
the cases involving “none”. For example, there is only case that does not include any girls, BBB. This is
the complement of “at least one”. Therefore, the complement of “at least one” is “none”. In this case
the probability of “none” is 1/8 and so the probability of “at least one” is
𝑃(𝐴̅) = 1.0 − 𝑃(𝐴) = 1.0 −
1 7
=
8 8
Expected Value
If you were to flip a coin ten times, how many heads would you expect to get? The probability of getting
a heads on any one flip is ½ so ten flips should result in half that number or five heads. We define
Expected Value of an event to be
𝑬 = 𝒏𝒑
where n is the number of trials and p is the probability of the event occurring.
Question #3
The probability of rolling a seven in dice is 1/6. Therefore, how many times would you expect a seven to
occur if you were to roll the dice 36 times.
There’s another very useful way to use expected value. Let’s say that your chances of winning a bet is
90%, and if you win you get $200, but if you lose, you lose $1000. Would you take this bet? The
expected return is given by the following formula,
𝑬 = 𝒑𝑾 − 𝒒𝑳
where p is the probability of winning, q is 1 – p, the probability of losing, W is the amount you win and L
is the amount you lose? The expected return in this case is
𝐸 = (. 90)(200) − (. 10)(1000) = 80
The expected return is positive, so this is a good bet, at least mathematically, but would you take this
bet? Many people would be reluctant to do so, because even a 10% chance of losing $1000 would be
too much for them. Suppose however, you were given enough seed money to cover 100 trials. Now
would you take the bet? This case is similar to speculating in the stock market. If you have a system
that has a winning edge, and if you have the resources to cover any losses early on, then as long as you
get many chances to repeat the bet on a mathematically winning game, many people will play the game.
This is the end of Unit 2.
Turn now to your MyMathLab homework to get more
practice with these concepts.
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