Examples
Ex 1:

A box contains 4 red and 6 blue balls. Another box contains 4 blue and 6 green balls. A ball is taken out from each box. Find the probability of one ball being red and the other green.

Sol:

Both the boxes have 10 balls each (of different colors though).
There are 4 red balls in the first box and 6 green balls in the second.
Let A be the event of taking out a red ball and B that of a green ball.
P(A) = 4/10 and P(B) = 6/10
The two events A and B are independent.
∴ P(A ∩ B) = P(A).P(B) = (4/10).(6/10) = 0.24

Ex 2:

Consider an experiment of drawing two cards at random from a pack of cards without replacement

Let us define the two events in the experiment as
i. Event A is getting a Jack in the first pick
ii. Event B is also getting a Jack in the second pick
Sol:

You should be knowing that a deck has 52 cards and that there will be 4 Jacks ( J ) in the deck.
The probability of occurrence of event A
P(A) = 4/52
Now the probability of occurrence of event B depends on whether event A is successful or not.
Let us analyze the two cases.

Case i: Event A is successful

Since a Jack is already drawn in the first pick, we are left with 51 cards and 3 Jacks in them.
⇒ Odds of picking second card as Jack = 3/51
∴ Probability of events A and B happening
P(A and B) = (4/52) × (3/51)

Case ii: Event A is unsuccessful

The probability of 'A' being unsuccessful = 1 – P(A) = 1 – (4/52) = 48/52
After the first pick, all the four Jacks will be there in the 51 cards
⇒ Odds of picking second card as Jack = 4/51
∴ Probability of events A and B happening
P(A and B) = (48/52) × (4/51)

The over all probability is the sum of the probabilities in the two cases

i.e, (4/52) × (3/51) + (48/52) × (4/51) = 204/2652

Let us modify the same experiment from "without replacement" of cards to with replacement of cards.

As earlier P(A) = 4/52
After drawing the first card, it is replaced back before drawing the second card.
So 4 Jacks remain immaterial of event A being successful or not.
⇒ P(B) = 4/52
∴ P(A and B) = (4/52) × (4/52) = 1/169
In this experiment, events A and B are said to be independent.

But are A and B exclusive events?

Events are said to be exclusive if occurrence of one event eliminates the occurrence of the second i.e, both the events can occur simultaneously.
We can therefore conclude:
  • Independent events are not mutually exclusive
  • Mutually exclusive events are not independent.
  • There is a sense of chronology in case of dependent events.
In the above example, A and B being independent events, they are also exclusive events.