12.2 Basic Concepts

If you roll a die, pick a card from deck of playing cards, or randomly select a person and observe their hair color, we are executing an experiment or procedure. In probability, we look at the likelihood of different outcomes. We begin with some terminology.

Probabilities are essentially fractions, and can be reduced to lower terms like fractions.

Example 3

Let's say you have a bag with 20 cherries, 14 sweet and 6 sour. If you pick a cherry at random, what is the probability that it will be sweet?

import random
from fractions import Fraction

random.seed(0)          # <-- CHANGE the 0 to re-roll every simulation below

SWEET, SOUR = 14, 6     # <-- CHANGE the bag and re-run
bag = ["sweet"] * SWEET + ["sour"] * SOUR

p_sweet = Fraction(SWEET, SWEET + SOUR)
print("bag: %d cherries, %d sweet" % (len(bag), SWEET))
print("P(sweet) = %d/%d = %s = %.4f" % (SWEET, len(bag), p_sweet, float(p_sweet)))
print("P(sour)  = 1 - P(sweet) = %s = %.4f   <- the COMPLEMENT"
      % (1 - p_sweet, 1 - float(p_sweet)))
print()

# Now stop trusting the formula and go pick cherries.
print("%12s %10s %12s %12s" % ("trials", "sweet", "proportion", "off by"))
print("-" * 50)
for trials in (10, 100, 1_000, 10_000, 100_000):     # <-- CHANGE this list
    hits = sum(1 for _ in range(trials) if random.choice(bag) == "sweet")
    prop = hits / trials
    print("%12s %10d %11.4f %+12.4f"
          % ("{:,}".format(trials), hits, prop, prop - float(p_sweet)))
print("-" * 50)
print("The proportion wanders toward %.4f. That is the law of large numbers:" % float(p_sweet))
print("probability is what the long-run frequency settles down to, not a promise")
print("about any single draw -- and not a promise that 10 draws give 7 sweet.")
print()

# The book's caution: this formula needs every outcome to be EQUALLY LIKELY.
# Suppose the sour cherries are bigger and come to hand twice as often:
weights = [1] * SWEET + [2] * SOUR      # <-- CHANGE the 2 to 1 to make it fair again
trials = 100_000
hits = sum(1 for _ in range(trials) if random.choices(bag, weights)[0] == "sweet")
print("If sour cherries are 2x easier to grab, 'P(sweet)' measured by grabbing is %.4f,"
      % (hits / trials))
print("not %.4f. Counting outcomes only works when they are equally likely." % float(p_sweet))

Show solution

There are 20 possible cherries that could be picked, so the number of possible outcomes is 20. Of these 20 possible outcomes, 14 are favorable (sweet), so the probability that the cherry will be sweet is $\frac{14}{20} = \frac{7}{10}$ .

There is one potential complication to this example, however. It must be assumed that the probability of picking any of the cherries is the same as the probability of picking any other. This wouldn't be true if (let us imagine) the sweet cherries are smaller than the sour ones. (The sour cherries would come to hand more readily when you sampled from the bag.) Let us keep in mind, therefore, that when we assess probabilities in terms of the ratio of favorable to all potential cases, we rely heavily on the assumption of equal probability for all outcomes.

Example 4

Compute the probability of randomly drawing one card from a deck and getting an Ace.

Show solution

There are 52 cards in the deck and 4 Aces so

$P (A c e) = \frac{4}{52} = \frac{1}{13} \approx 0.0769$

We can also think of probabilities as percents: There is a 7.69% chance that a randomly selected card will be an Ace.

Notice that the smallest possible probability is 0 – if there are no outcomes that correspond with the event. The largest possible probability is 1 – if all possible outcomes correspond with the event.

In the course of this chapter, if you compute a probability and get an answer that is negative or greater than 1, you have made a mistake and should check your work.

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.