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Maths > Probability > Independent Events

Independent Events

Suppose you have a pack of 52 well-shuffled cards. You ask one of your friends to draw any of the cards from this pack. After the first choice, you ask him to draw another one. He does accordingly. Are the results of the two draws the same? Do you get the same card in each draw? Are the chosen independent of each other? It depends on your choice whether you replace the first drawn card or not. This gives rise to a new concept of the independent event.

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Independent Event

The literal meaning of Independent Events is the events which occur freely of each other. The events are independent of each other. In other words, the occurrence of one event does not affect the occurrence of the other. The probability of occurring of the two events are independent of each other.

An event A is said to be independent of another event B if the probability of occurrence of one of them is not affected by the occurrence of the other.

Suppose if we draw two cards from a pack of cards one after the other. The results of the two draws are independent if the cards are drawn with replacement i.e., the first card is put back into the pack before the second draw. If the cards are not replaced then the events of drawing the cards are not independent.

Statistically, An event A is said to be independent of another event B, if the conditional probability of A given B, i.e, P(A | B) is equal to the unconditional probability of A. P(B) ≠ 0.

P(A | B) = P(A)

The term mutually exclusive should not be mixed with the term independent. The term mutually exclusive is related to the occurrence of an event. By independence of events, we mean the independence of probability of occurrence of events.

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Theorem 1

The events A and B are independent if P(A ∩ B) = P(A) P(B).

Proof: From the definition of an independent event, we have P(A | B) = P(A) ⇒ P(A ∩ B) ⁄ P(B) = P(A)

or, P(A ∩ B) = P(A) P(B). Here, P(B) ≠ 0.


For two events A and B such that P(A) ≠ 0, P(B) ≠ 0. If A is independent of B, then B is independent of A.

Proof: If A is independent of B, we have

P(A | B) = P(A) or, P(A ∩ B) / P(B) = P(A) or, P(A ∩ B) = P(A) P(B) … (I)

P(B|A) = P(A ∩ B) / P(A) = [P(A) P(B)] ⁄ P(A) = P(B) [from I]

So B is also independent of A.

Theorem 3

If A and B are independent events, then the events A and B’ are also independent.

Proof: The events A and B are independent, so, P(A ∩ B) = P(A) P(B).

independent event

From the Venn diagram, we see that the events A ∩ B and A ∩ B’ are mutually exclusive and together they form the event A.

A = ( A ∩ B) ∪ (A ∩ B’).
Also, P(A) = P[(A ∩ B) ∪ (A ∩ B’)].
or, P(A) = P(A ∩ B) + P(A ∩ B’).
or, P(A) = P(A) P(B) + P(A ∩ B’)
or, P(A inter B’) = P(A) − P(A) P(B) = P(A) (1 – P(B)) = P(A) P(B’)

Mutually Independent Events

Three events A, B, and C are mutually independent if

P(A ∩ B) = P(A) P(B)
P(B ∩ C) = P(B) P(C)
P(A ∩ C) = P(A) P(C)
P(A ∩ B ∩ C) = P(A) P(B) P(C)

Solved Example for You

Question 1: Let A and B are two independent events such that P(A) = 0.2 and P(B) = 0.8. Find P(A and B), P(A or B), P(B not A), and  P(neither A nor B).

Answer : Given P(A) = 0.2 and P(B) = 0.8 and events A and B are independent of each other.

P(A and B) = P( A ∩ B) = P(A) P(B) = 0.2 × 0.8 = 0.16.

P(A or B) = P(A ∪ B) = P(A) + P(B) – P(A ∩ B) = 0.2 + 0.8 – 0.16 = 0.84.

P(B not A) = P(B ∩ A’) = P(B) – P(A ∩ B) = 0.8 – 0.16 = 0.64.

And  P(neither A nor B) = P(A’ ∩ B’) = 1 – P(A ∪ B) = 1 – 0.84 = 0.16.

This concludes our discussion on the topic of the probability of an independent event.

Question 3: What is an example of an independent event?

Answer: Two events, X and Y, are independent if X occurs won’t impact the probability of Y occurring. More examples of independent events are when a coin lands on heads after a toss and when we roll a 5 on a single 6-sided die. Then, when selecting a marble from a jar and the coin lands on the head after a toss.

Question 4: What are independent and dependent events?

Answer: An independent event refers to an event where the result does not get impacted by some other event. On the other hand, a dependent event is one which does get affected by the result of a second event

Question 5: What is the multiplication rule for independent events?

Answer: The multiplication rule for independent events is said to relate the probabilities of two events to the probability that they both happen. In order to make use of the rule, you need to be having the probabilities of each of the independent events.

Question 6: What does it mean for an event to be independent?

Answer: When we say two events are independent of each other, we mean that the probability that one event will occur in no way will impact the probability of the other event that is taking place. For instance, two independent events will be when you are rolling a dice and flipping a coin.

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