Dependent and Independent Events in Probability

In this class, We discuss Dependent and Independent Events in Probability.

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The reader should have prior knowledge of mutually exclusive events. Click Here.

Dependent Events:

Two events A and B, are said to be dependent if occurring of event A will affect the probability of event B.

Example:

A bag contains five black and six green balls.

Randomly pick a ball fom the bag two times.

A = First time, should pick a black ball

B = Second time, should pick a black ball

Probability to pick a black ball the first time = 5c1/11c1

The probability of selecting a black ball a second time depends on the first-time selection.

First attempt if we select green ball. Probability of selecting black ball second time = 5c1/10c1

First attempt if we select black ball. Probability of selecting black ball second time = 4c1/10c1

Our conditional probability class discussed how to find the probability of dependent events?

After selecting a ball the first time in the above random experiment, we are not replacing the ball in the bag.

We are selecting with replacement.

Independent Events:

First time probability of picking black ball = 5c1/11c1

Now we replace the ball in the bag.

Second-time probability to pick a black ball = 5c1/11c1

Whatever ball we pick the first time, with replacement, the probability of a second attempt is not getting affected.

We call these types of events independent events.

Independent Events:

Two events, A and B, are said to be independent events.

The occurrence of one event will not affect the probability of another event.

Another Example:

Toss a coin two times.

A = First time should toss head.

B = Second time should toss head.

When we toss the first-time coin probability of getting a head = 1/2

When we toss the coin a second time. probability of getting a head = 1/2

The first event will not affect the probability of the second event.

Independent Events:

P(A ∩ B) = P(A) P(B)

P(A ∩ B) = occurring of both the events.

From the random experiment, toss a coin two times. P(A ∩ B) = 1/2 * 1/2 = 1/4

Proof:

Toss the coin two times.

P(Both Heads)?

Sample Space S = {HH, HT, TH, TT}

Event E = {HH}

P(E) = 1/4.