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Probability is a mathematical measure of the likelihood of a particular event occurring.
Numerically, we give probabilities a value between zero for absolutely impossible and
one for absolutely certain. We often describe these values as percentages: zero percent
for "will never happen", and one hundred percent for "certain to happen".
Of course, as well as numbers we often describe probabilities using other words, like "fifty-fifty"
(which bears an obvious relationship to fifty percent), "possible", "likely", "unlikely",
"probable", "most likely", even words like "perhaps", "might", "maybe", "could", "usually",
"generally" ... The exact placement of many of these words along this scale is somewhat
ambiguous, and to be honest that's why we need to describe probability using numbers
instead!
Experimental probability is measured by calculating the relative frequency of a certain event
occurring. Experimental probability is based on actual experimental outcomes: you try the
experiment many times, and you calculate the relative number of times that the outcome
you are interested in occurs.
For example, if Ben is a pistol shooter who hits the target seventy-two times out of eighty
shots, then the experimental probability of Ben hitting his target is defined as the relative
frequency: the number of on-target shots, seventy-two, out of the total number of shots,
which is eighty. That's nine-tenths of the time, or ninety percent probability—or as
a number between zero and one, it's point nine.
Or, suppose I have a bag full of coloured marbles. I'm going to pull one marble out
of the bag at a time, and put each one back and shake up the bag before pulling out the
next. If I do that fifty times, and I get a green marble thirteen times out of those
fifty draws, then what's the experimental probability of getting a green marble? Well,
it's the relative frequency: the number of times I got a green marble over the total
number of times I tried. Thirteen out of fifty is twenty-six percent, or zero point two six.
You can express it any of these three ways.
Notice that the experimental probability has nothing to do with the actual number of marbles
in the bag, green or otherwise. Experimental probability is based only on the outcome of
doing the experiment many times. For experimental probability, you have to first do the experiment.
You can even calculate experimental probability from a graph of the data. Here's a histogram
of the results of a survey of houses in a particular suburb. If you were to choose a
house in this suburb at random, what would be the experimental probability that the house
you chose has at least four people living there? Well, the experimental probability
is the relative frequency of the event of interest. Here are the houses with "at least
four" people: two outcomes, with a total frequency of fifteen plus eight, or twenty-three. The
"number of trials" is the total number of houses surveyed, which is the sum of all the
frequencies in the graph: one, plus four is five, plus ten is fifteen, plus fifteen is
thirty, plus eight makes thirty-eight. So the relative frequency is twenty-three over
thirty-eight, which is about sixty point five percent.
You can use the relative frequency to predict
the number of times a particular event will occur for a different number of trials. This
is called the expected frequency. You just use the probability as a proportion of the
new number of trials. Notice that this is really just a simple percentage calculation:
The probability is the percentage of the time that you expect this particular outcome to
occur. Let me show you.
Let's go back to our first example, Ben the pistol shooter who hits the target seventy-two
times from eighty shots. We decided that the relative frequency was seventy-two out of
eighty, or ninety percent. So if on another day he shoots twenty times, how many times
would you expect Ben to hit his target?
The expected frequency is the probability multiplied by the number of trials. That's
ninety percent of twenty trials, which is eighteen times. Easy enough?
Or our second example, the bag of coloured marbles. I got a green marble thirteen times
from fifty draws, so the relative frequency was thirteen out of fifty. How many green
marbles should I expect to get if I make two hundred and twenty draws?
The expected frequency is the probability multiplied by the number of trials. That's
thirteen over fifty, times two hundred and twenty. You might want a calculator. I need
to round off my answer here, to make sense in the context of the question: I would expect
to pull out a green marble around fifty-seven times.