A RANDOM REAL FROM A RANGE
Select : Random Real [Lower, Upper]
Select : Random Integer [Lower,Upper]
A RANDOM SELECTION OF VALUES
Select : Random Sample (k items)
A RANDOM INTEGER FROM A RANGE
A RANDOM INTEGER BETWEEN 0 & YOUR CHOICE
Select : Random Integers [0. Upper]
A RANDOM REAL BETWEEN 0 & YOUR CHOICE
Select : Random Real [0.0, Upper]
A RANDOM REAL FROM 0.0 to 1.0
Select : Random Real [0.0,1.0]
A RANDOM 0 or 1
Select : Random 0 or 1]
A COIN FLIP
Select : Coin Flip]
A RANDOM ROLL OF DICE
Select : Dice Rolls]
Probability and random events go hand in hand. See the selection of vCalc equation to the right that generate random choices.
Flickr / Author: Paul Hudson - Dice
Probability measures the likelihood that an event will occur. The probability of an event occurring lies somewhere between the impossible and the certain.
Probability is represented mathematically as a number between 0 and 1. The closer to 1 is the probability of an event, the more certain it is that the event will occur.
The tossing of a coin is a simple example of probability in action. For each flip of the coin there are two possible outcomes: "heads" or "tails". Click the link at the right to A COIN FLIP, to try it yourself. Each time you flip (click the reset of the equation) you will generate a "heads" or a "tails". If you do this enough times and count the heads and tails they should come out even. The number of flips over time will come out very,very close to 50/50 -- depending on how many times you flip.
If the coin is "fair", both "heads" or "tails" are equally probable, since no other outcomes are possible, the probability is 50% for flipping a "heads" and 50% for flipping a "tails". The probability value for either possible outcome is expressed as 0.5.
The probability of either possible outcome is also said to be 1 out of 2. Try it HERE.
These concepts of probability define the basis of formal definitions of probability theory. Probability as a sub-heading of Statistics is applied in almost every scientific and mathematic field of study study. This same basic concept of the probability of occurrence of an event within a set of possible events is the basis of theory defining gambling, weather studies, quantum physics, insurance, banking, computer learning, game theory, and even philosophy.
NOTE: many vCalc equations are embedded throughout vCalc descriptive pages like this page. Even though they may not stand out in the text, if you hover over the name of an equation it will likely be linked to an actual, pop-up executable equation. For example: Arithmetic Mode
Fundamental probability theory derives from the simplest formula:
Eq 1: `p(A) = n_A/N` Try it HERE
This is the equation defining probability as the ratio of the number of events belonging to the subset, A, to the total number of possible events. N is the total number of possible events.
If we have five numbered balls, picking any one of the five balls at random has `N`= 5 total possibilities. Three of the balls have odd numbers on them, so the subset A of N that represents the possibilities that one of the odd-numbered balls will be picked randomly is three. Thus the probability of picking randomly one of the odd balls is `p(A) = 3/5 = 0.6` or 3 out of 5.
We think of basic probability as identifying how many possibilities might occur within all possible cases (or states or conditions or events).
The addition rule states: in a set of mutually exclusive random events the probability of occurrence of either one event or another event is the sum of their individual probabilities. The rule is given as:
Eq 2: `p(A or B) = p(A) + p(B)` Try it HERE
Selecting a single card, such as a 10 of hearts, from a shuffled deck has a probability of `p(A) = 1/52`, 1 out of 52. Picking some other specific card, such as a king of clubs likewise has a probability of `p(B) = 1/52`. The probability then that the randomly chosen card will be either a 10 of hearts or a king of clubs is:
`p(A or B) = 1/52 +1/52 = 1/26`
We think of mutually exclusive probability as two events that might occur separately, one occurs or the other occurs..
The multiplication rule states: the probability of two or more independent events occurring on separate occasions is the product of their probabilities.
Eq 3: `p(A,B) = p(A) * p(B)`, where `p(A,B)` is read as the probability of both A and B occurring. Try the calculation HERE.
Flickr / Author:ICMA Photos - Coin Toss
If we toss a coin twice, what is the probability that the first flip will produce a heads and the second flip will produce a tails. The probability of a heads turning up on a single flip is `p(A) = 1/2` and the probability of a tails turning up on a single flip is `p(B) = 1/2`. The Multiplication Rule then tells us the probability of a heads occurring on one flip and a tails occurring on another flip is `p(A,B) = p(A) * p(B) = 1/2 * 1/2 =1/4`
We think of independent events as two or more events which have starting conditions that don't affect each other, the starting conditions define the probability of one event and the other event's starting conditions define the other event's probability.
Conditional probability is defined: the likelihood that an event will occur, given the fact that another event or series of events has already happened.
The probability of B given A is written as `p(B|A)`, where p(B|A) = (p(A,B)) / (P(A)). Try it HERE
Suppose we have a 100 balls and draw one randomly which has the number 5. The probability that we would draw the ball with the number 5 is `1/100`. We know that from the Definition of Probability above.
We replace the ball numbered 5 into the set so we can next draw again from the same 100 balls. This time we draw a ball numbered 38 from the set of 100 balls. So what is the probability that the ball we draw this time has 38 on it? Yes, you guessed it: `p(B) = p(B|A) = 1/100`.
The probability remains the same as long as we replace the ball in the set and randomly draw another ball from the same original 100. Replacing the ball so the same set is randomly drawn from each time makes the two draws independent events.
We think of independent but conditionally related events as multiple events that have the same probability, the same probability they would always have to occur.
Now suppose we draw the 5 from the same set of 100 balls described in the previous example. Again the probability of drawing the number 5 from a set of 100 is `1/00`. But this time we don't add the number 5 ball back into the set of 100, so now we will select the second ball from the set of 99 balls without the number 5. What is the probability that the first ball draw is a 5 and the second ball is an 11?
We define here the probability that event A is followed by event B and A and B are not independent.
The probability of the first draw is p(A) and the probability of the second draw is p(B|A) (read probability of B given A).
Since the probabilities of a series of events is subject to the multiplication rule, the joint probability of this series of events is the product of the individual events' probabilities.
`p(A, B) = p(A) * p(B|A) = 1/100 * 1/99 = 1/(9,900)` Try it HERE
How many times have you seen or heard someone suggest that a random event is "due" to occur. An ignorant gambler flipping a coin will sometimes remark that because quite a number of heads have come up in a row that a heads is "due". The fact is that each and every flip of the coin has precisely the same probability of turning up a heads. The random flip could quite literally result in 1000 straight heads and the probability of the very next flip producing a tails will again be exactly 1/2.
Similarly, people who play the Powerball every week believing their chance will improve over time are essentially, unmistakably wrong. The probability resets each time the numbers are drawn and next week the probability is the same. CLICK HERE to see the embarrassingly minuscule probability Powerball players are paying out money weekly for.