Independence of Events: Definition, Checking for


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What is Independence of Events?

independence of events
A die roll is an independent event.

Independence of events means that previous events have no effect on following events. For example, you roll a die once and get a 3; the next roll you get a 4. Neither roll has an effect on each other; they are independent.


Sometimes, it may seem like there is independence of events when in fact there is not. A loaded die, an unfair coin, or lottery balls that have more ink on them (which makes them heavier, and more likely to drop). Card games like poker, also do not have independence of events; the probability of being dealt a particular card changes as the cards are dealt. With a standard deck of cards, your probability of getting an ace is 4/52 (there are four aces, and the standard deck has 52 cards). But if you have already been dealt one card that isn’t an ace, there are 51 cards remaining, which lowers your probability of getting an ace to 4/52.


Checking for Independence of Events

There are four formulas for checking for independence of events:

  1. P(B | A) = P(B)
  2. P(A | B) = P(A)
  3. P(B | A) = P(B | not A)
  4. P(A and B) = P(A) * P(B)

For example, the probability of getting a heads on a coin flip (we’ll call this event A) is 50% or 0.5. And the probability of a tails (we’ll call this event B) is also 0.5.

Condition 1: P(B | A) = P(B). In English, you would read the left hand side of this equation as “the probability of event B happening, given that event A has happened.” This statement should equal the probability of B.
We know the probability of event B (tails) is 50%, so we need to figure out the left hand side of the equation to compare it. Given that we have already had event A (heads), what is our probability of getting tails? Logic should tell you the probability stays the same: 50%. You could run an experiment by flipping a coin a few thousand times to show that this fact is true, but you don’t need to because of this handy formula.

You don’t actually have to work your way down the list.
If any of these conditions hold, events are independent. That’s because all four equalities are, mathematically speaking, equivalent. Therefore, you only need to check one of these conditions to show that events are independent [1]. The other conditions are useful for figuring out independence if you have other snippets of information about events. For example, you might know the probability of two events happening together (A and B) or you might know the probability of an event not happening (e.g., “not A”).


References

[1] Conditional Probability and Independence.

CITE THIS AS:
Stephanie Glen. "Independence of Events: Definition, Checking for" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/independence-of-events/
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