1 is the probability of an event that is sure to happen. The probability that the sun will rise tomorrow is 1.
0 is the probability of an event that cannot happen. The probability that a Bridge hand will contain a Joker is 0, because Bridge is played with a deck from which the Jokers have been removed.
Here's how you get probabilities between 0 and 1: Suppose there are a number of equally likely events. Some events represent a "win" and some a "loss." The probability of winning is the number of winning events divided by the total number of events. Such well-defined situations arise in practice only in games.
For example, consider a coin with a "heads" side and a "tails" side. If we assume that the two sides are equally likely to be up when the coin is tossed and lands on a table, then the probability of "heads" is 1/2 and the probability of "tails" is 1/2.For another example, an American roulette wheel has 38 spaces. Each space has a number and a color. Eighteen of the spaces are red. Eighteen are black. Two are green. If you bet that the ball will fall in a red space, your probability of winning is 18/38.
Your turn:
A deck of playing cards has 52 different cards, if we remove the Jokers. I draw a card at random from that deck. What is the probability that the card will be the Queen of Diamonds?
Some examples:
What is the mean of these numbers: 1, 1, 2, 8?
The mean is a measure of central tendency. The mean height of 21-year-old men in the U.S. is 177 cm, or 5' 10". The mean height of women here is 164 cm, or 5' 4". This tells us that, in the U.S., men are generally taller than women. It does not imply that every man is taller than every woman.
For the coin, the expected value is:
(0 times the probability of 0) + (1 times the probability of 1).
If the coin is fair, so that the two sides are equally likely to come
up, then the expected value is:
(0 times 1/2) + (1 times 1/2) = 0 + 1/2 = 1/2.
If all outcomes are equally likely, then the expected value will equal the mean of all the outcomes.
Suppose we flip our coin many times, and calculate the mean of all the outcomes of the tosses. The more flips we do, the more likely it is that the mean of all the accumulated outcomes will be close to the expected value. The simulation above does this. If you say that a head is worth 1 and a tail is worth 0, the "proportion of heads" fraction is the mean of all the outcomes so far. It does tend to get closer and closer to 0.5.
General instructions
for calculating a variance: |
For example, suppose you have
a coin with 0 and 1 painted on its sides: |
Take each data item. | The data items are 0 and 1. |
Subtract the mean from each. | Subtract the mean ( 1/2 ) from each: -1/2 and 1/2 |
Square each. | Square each: 1/4 and 1/4 |
Add them all up | Add them all up: 1/4 + 1/4 = 1/2 |
Divide by the number of data items
to get the variance. |
Divide 1/2 by the number of data items ( 2 ), to get 1/4.
The variance is 1/4. |
Your turn:
What is the variance of these numbers: 3, 3, 3, 3. (Yes, the four numbers are the same.)
This one is harder:
What is the variance of 1, 1, 2, 8?
Notice that 3, 3, 3, 3 and 1, 1, 2, 8 have the same mean but different variances. If you were designing a system for handling eggs, you would want to know the both mean and the variance of the size of the eggs.
If the outcome numbers are continuous, meaning that the outcome could be any number in either a finite range or an infinite range, then you can't list all possible outcomes and each one's probability. What you can do instead is give a mathematical formula that would tell you, for any given x , what the probability is that the outcome will be less than x . This is the distribution function. Distributions are often visualized using the density function. On a graph of the density function, if you pick any two possible outcome values on the x axis, and draw vertical lines at those values up from the x axis to where they intersect the density function's curve, then you will have an enclosed area with straight sides, a straight botton, and a curved top. The size of that area tells you the probability that the outcome will be between the two x values.
For example, a distribution we'll be using often this semester is the
normal
distrbution. Its density function is "the bell curve."
This is the density function for a normal distribution with a mean
of 0 and a standard deviation of 1. The density function is constructed
so that the total area under the curve is 1. (The area under the curve
is unbounded, because the curve goes off forever in both directions without
ever quite touching the x axis, but even so the area under the curve
is finite and equal to 1.) The area of the shaded region is the probability
of a outcome number between 1 and 2.
For our coin with a 0 on one side and a 1 on the other, the data items are x_{1} = 0 and x_{2} = 1. N = 2.
The course readings use this notation often. If it seems intimidating, take the time to work through what the expression means. Proceed methodically and you can do it.