Probability: Dependent or Independent Events?

Question

I understand that the probability of tossing a coin and it landing or either heads or tails is 50% and that flipping it again is an "independent event" and does not change the probability at all. That is, the chances of the coin landing on heads or tails is still 50% regardless of whether it landed on heads or tails the first time.

However, what about something like gasoline prices? Today, gasoline prices can either be up or down (50%). Similarly, tomorrow gasoline prices can either be up or down (50%). But what if gasoline prices were down today and a study of past gasoline price patterns ("empirical probability" from experimenting) demonstrates that if they are down today, they also have been down the following day (60%) of the time--a ten (10%) difference? Is whether gasoline prices were up or down today an "independent event" from whether they will be up or down tomorrow, or a "dependent event"? I'm looking for some definitional clarity here.

Answer 1

For gas price data you will see a strong correlation between adjacent daily values, and a much smaller but definite weekly (5 day) correlation also.

Take the daily price data and auto-correlate it. If the data values are mutually independent, the auto-correlation will be an impulse (or close to it). If you get anything else (like a bell shaped curve), then the data points are dependent. The more dependent, the more unlike an impulse the auto-correlation will be.

let s(i) be the daily price on day "i" for some defined range of "i".

Then the auto-correlation will be

a(j) = K * sum(i = min_i to max_i - j, s(i) * s(j+i) )
K = 1/((max_i - min_i - j)*a(0))

an impulse is
a(j) = 1 for j=0 and 0 for any other value of j.

One can perform a spectral analysis of the auto-correlation data to find important features in the data that can be used for prediction.

See
http://www.geocities.com/derekcowley/public/correlation.pdf
definitely not an impulse.

calculation in
http://www.geocities.com/derekcowley/public/correlation.xls

Answer 2

Flipping a coin and the change in gas prices aren't exactly the same thing. While both sides of the coin are equally likely to be shown after a flip. Predicting gas prices is similar to predicting the weather. It is not completely random. For an example, if you experienced sunny days all week, it is highly unlikely that it will rain the next day. It is called conditional probability. Because the events are not independent you have to condition upon passed events. This is similar to gas prices. They are not purely independent of everything else. These days gas prices are really high and the probability that they will be low tomorrow are extremely low.

Answer 3

Definitely dependent events. Today's gasoline price compared to yesterday's gasoline price are dependent--although at first blush, it might seem like there is a 50-50 probability that prices will be up or down on any given day. First, think about it anecdotally and forget statistics for a moment. Generally, prices tend to trend, both up and down, for myriad reasons, whether they be grain prices, stock indices, individual stocks, gasoline, crude oil etc. Gasoline prices, for instance, go up and down based upon supply and demand, seasonal factors, mass psychology, the actions of speculators in the futures markets etc. That's where the sayings "higher prices beget higher prices" and "lower prices beget lower prices" come from.

Now, back to statistics and your example. If you were to graph all of the days where gasoline prices were up versus down, it's likely to be around 50-50 of the total number of days that you are sampling. However, I am near certain that if you then count and plot the number of days where gasoline prices were up two days in a row, the number will be smaller. Further, if you then take the number of days where gasoline prices were up three days in a row, the number will be smaller yet. By the time you count and plot the number of times it was up, say, 13 days in a row, the number will become very small. This is likely to show up as one half of a bell curve. If you then did the same experiment with consecutive down days, it will most certainly form something close to the other side of the bell curve. As alluded to above by another poster, this will tell you that there is some correlation between the closing price of gasoline each day to the closing price of yesterday or the past 13 days etc. Inside of those prices, say 3.00 a gallon, is a wealth of information that none of us can readily see, but it really doesn't matter what the information is necessarily, because price is the ultimate arbiter of supply and demand, mass psychology etc. Over long periods of time (decades in many cases), if your sample is that large, you will be able to see cycles and price patterns the are emerge--largely because human nature, and things like fear and greed never really change--all of which are built into prices of things.

Answer 4

I believe Intel_Knight is correct.
The events are dependent.

If you randomly choose one day out of the year, there is a 50% probability that the gas prices were down on the day you choose; and there is a 50% probability that the gas prices were up.

If you randomly choose one day out of the year, where the gas prices on the previous day were up; then the probability of the prices being up on the day you choose is no longer 50%. Because the probability changes, you know that the two events must be dependent.

Use this test:
What is the probability of getting heads? 50%
What is the probability of getting heads, given the previous flilp was heads? 50%

There is no change so the events are independent.

What is the probability of prices being up? 50%
What is the probability of prices being up, given that the previous day there were up? 60%
The answers are different so there must be some dependence.

Answer 5

your gasoline prices example is correct - these events are not independent. They do that kind of forecasting a lot for company stocks.

Events are independent if joint probability can be computed by multiplying the probabilities of individual events that do not condition on what happened yesterday.

Answer 6

1

Answer 7

Its an independant event and it depends on the market, the availabilty, and the suppl amount. In other words there are several determing factors based on supply and demand.

Answer 8

It is dependent. I think if you pay attention on the role of human in this case, you will get the point. Math cannot explain cases that human has influence on them , well ; because human may do anything. I think psychology can explain such a case better.

Answer 9

1) Simply multiply the two %s together. i.e. When Rocco is placed in the trial, 60% of the time he will smell the treat, of those times he smells the treat he will get through the maze 85% of the time - so 60% x 85% = 51%. The chance that Biff will get the treat is 52.5%