Dependent Events

How to find the probability of dependent events: formula, example, and its solution.

Formula

Dependent events are the events
that do affect each other.

If A and B are dependent events,

then P(A and B) can be found
by multiplying P(A) and P(B'):

P(A and B) = P(A)⋅P(B').

P(A and B): Probability of both A and B happening
P(A): Probability of A happening
P(B'): Probability of B happening affected by A

You can see that
P(B') part is different
from the independent events formula.

So you should keep attention
when finding P(B').

Independent events

Example

After the first pick,
the picked number is not replaced.

So the first pick do affects the second pick.

So these two events are dependent events.

So find P(A),
find P(B'), the probability affected by A,
then find P(A and B)
by multiplying the probabilities.

First see the first pick.

Numbers from 1 to 10 are given.

So there are 10 numbers
that can be picked.

So n(S) = 10.

Set the event A as
picking the multiples of 3 at the first pick.

The multiples of 3 from 1 to 10 are
{3, 6, 9}.

So n(A) = 3.

n(S) = 10
n(A) = 3

So P(A) = 3/10.

Probability

Next, see the second pick.

The picked number is not replaced.

So there are, 10 - 1, 9 numbers
that can be picked.

So n(S'), the changed space sample, is 9.

Set the event B as
picking the multiples of 3 at the second pick.

One of the multiples of 3 is already picked
and not replaced.

So the remaning multiples of 3 are
{6, 9} or {3, 9} or {3, 6}.

So n(B') = 2.

n(S') = 9
n(B') = 2

So P(B') = 2/9.

It's good to compare this
with the independent events problem.

Because of no replacement,
P(B') part has changed.

Independent events - Example 3

P(A) = 3/10
P(B') = 2/9

So P(A and B) = (3/10)⋅(2/9).

Cancel 3 and reduce 9 to 3.

Cancel 2 and reduce 10 to 5.

Then (3/10)⋅(2/9) = (1/5)⋅(1/3).