# 4.20 Random Processes

Many physical phenomena are repeated or cyclic. Examples of periodic processes include waves, vibrations, oscillations and orbits. Several phenomena that are random in nature occur on fantastically different scales. Think of the difference between the random decay of a radioactive atom and the random impact of space debris on the Earth! A random process cannot be understood without dealing with the notion of probability. Random implies unpredictable. We are not sure of the outcome of a particular random event; we can only assign a probability to the outcome. However, while a single random event may be unpredictable, the sum of a large number of random events may be very well behaved and predictable.

Imagine an experiment where a measurement is made to place a card halfway through a deck of cards. The result is not perfect but it is rare for the card to be more than three or four places from the central position. Now suppose you are blindfolded and have to insert the card randomly. The likely outcome many separate placements is a random distribution. In other words, the card has an equal probability of appearing at any position in the deck from first to last. In mathematical shorthand we would say:

p_{n} = 1 / n

In this little equation pn is the probability that the card will appear any one of n different positions in the deck. Since a deck has 52 positions, p_{n} = 1/52 = 0.019. Each of the 52 positions has about a 2% probability of being selected. The probability is the same each time you repeat the random placement. For example, suppose your first placement was position 32. The next placement has a probability of 1/52 of being position 32 (or any other position) — the outcome of a random process does not depend on what came before.

Another random process is the act of tossing a coin. Since there are two possible outcomes, the probability of either heads or tails is p_{n} = 1/2. The next time you toss the coin there is still an equal 50% probability of either outcome. In the case of throwing a dice, p_{n} = 1/6. Once again, each successive throw of the dice has a 1/6 chance of turning up any particular number. Suppose you just tossed a coin five times and it came up heads every time? Doesn't that make it more likely that the next toss will be tails? No! The odds are still 50% because each event is independent of the others. Suppose you rolled a die hoping for a six and after 25 rolls you still had not seen a six. Doesn't that make is more likely that the next roll will be a six? No! The odds are still 1 in 6 for each roll. It seems counter-intuitive (and Las Vegas makes a lot of money of ignorance of statistics!) but the individual event is unpredictable regardless of the recent history.

Notice that in each case of a random experiment — random card placement, coin tossing, dice rolling — the result of a single event cannot be predicted with certainty. The outcome each time is random. However, the result of a large number of events is well behaved and predictable. We know that after tossing a coin many times the ratio of heads to tails will be close to one. (Try it!) In other words, a large number of experiments allows us to accurately measure that p_{n} = 1/2. A large number of rolls of the dice will confirm that we roll a six with a probability p_{n}= 1/6.

A deck of cards and pulling one card from it is a great example of a random process. . Click here for original source URL.

Notice something else about the experiment of randomly placing cards in a deck. Even though we placed the card 40 times, the card did not go into 40 different positions with 12 left over. Some positions got selected twice or even three times, others not at all. If there are equal numbers of each outcome, we call it a uniform distribution. A random distribution is not the same as a uniform distribution.

What is a physical example of a random process? Radioactivity is a process that is random in time. If we could stare at a single radioactive atom, we would not know exactly when it would decay. However, after a time equal to the half-life, the atom would have a 50% chance of having decayed. The familiar analogy is with a pan of popcorn cooking. If you stared at any one kernel, you could not predict when it would pop, but the time it takes for half the kernels to pop is a well-determined number that would be repeatable if you cooked many pans of popcorn the same way. We can write this:

p_{t} (1 half-life) = 1/2

The subscript "t" shows that our event is random in time. This probability applies to a single atom but it applies equally to a large number of atoms. After a half-life has passed, half of a very large number of radioactive atoms will have decayed. After twice the amount of time, p_{t} (2 half-lives) = 3/4. We know from our discussion of radioactivity that 75% of the atoms will have decayed after two half-lives.

Astronomers believe that the arrival of large objects that impact the Earth is essentially random. We can make a clear distinction between a random and a periodic process by using an ordered deck of cards as an example of a periodic process. If you turn the cards over one by one, a Jack (or a 4 or any value of card) will turn up every 13th card. This sequence is completely predictable. Now suppose turn over the cards of a well-shuffled deck. Where in the sequence will the Jacks turn up? There is no way to tell. There is no fixed spacing or periodicity of the Jacks; their position in the sequence is completely random.

The impacts of large pieces of space debris that cause mass extinctions may well be random (think of the occurrence of Jacks in a well-shuffled deck of cards as impacts and the sequence from 1 to 52 as a time sequence). Studying the fossil record might tell us how many large impacts occur over the past few billion years. Let's say this analysis implies that there is a 50% probability of an impact within any 100-million-year period. To use math terminology, p_{t}(100 million years) = 1/2. This probability gives the impact rate on average, but it gives us no idea exactly when the next impact will occur. The arrival rate of doom and gloom from the skies is random and cannot be predicted.