# Lesson: Independent Events

## Comment on Independent Events

### A={2,3,4,5}

A={2,3,4,5}
B={4,5,6,7,8}

Two integers will be randomly selected from the sets
above, one integer from set A and one integer from
set B. What is the probability that the sum of the two
integers will equal 9 ?

Hi can u pls explain this que. Im getting confused

### The best/fastest way to solve

The best/fastest way to solve this question is to apply the basic probability formula.

So, P(sum is 9) = (number of pairs that add to 9)/(total number of pairs)

We can select a number from set A in 4 ways, and we can select a number from set B in 5 ways. So, the TOTAL number of pairs = 4 x 5 = 20

Now the numerator: number of pairs that add to 9
Let's list all possible pair:
- select 2 from set A, and 7 from set B
- select 3 from set A, and 6 from set B
- select 4 from set A, and 5 from set B
- select 5 from set A, and 4 from set B
So, there are 4 pairs that add to 9

So, P(sum is 9) = 4/20 = 1/5

### Thanks Brent. Understood now

Thanks Brent. Understood now.

### Hi Brent,

Hi Brent,
How do I calculate the first example (selecting balls without replacement) if what I wanted to know was the probability of the second ball being green?

I mean, I understand that the two events are dependent, however, how can I know what is the probability of event B occurring if I don't know whether or not event A occurred.

What I got so far is:
If the first ball is green, then the probability of the second ball being green is: P(B|A) = 3/6 = 1/3

If the first ball is not green, then the probability of the second ball being green is: P(B|A) = 4/6 = 2/3

How do I reconcile these two probabilities into one?

### Sorry, I'm not 100% sure what

Sorry, I'm not 100% sure what probability you are trying to calculate.

If you want to calculate P(2nd ball is green), then we can say...

P(2nd ball is green) = P(1st ball is green and 2nd ball is green OR 1st ball is red and 2nd ball is green)

= P(1st ball is green AND 2nd ball is green) + P(1st ball is red AND 2nd ball is green)

= [P(1st ball is green) x P(2nd ball is green)] + [P(1st ball is red) x P(2nd ball is green)]

= [4/7 x 3/6] + [3/7 x 4/6]

= 12/42 + 12/42

= 24/42

= 4/7

ASIDE: Notice that P(2nd ball is green) is equal to P(1st ball is green). For more on this phenomenon see my post (7 posts down) at: http://www.beatthegmat.com/beat-this-probability-qs-t185719.html

Does that help?

### Hi Brent,

Hi Brent,
What I want to calculate is the probability of the second ball being green (similarly to the question on the post you mentioned).

It's clear now. I see where I went wrong.
Thanks

Gordon buys 5 dolls for his 5 nieces. The gifts include two identical S beach dolls, one E, one G, one T doll. If the youngest niece doesn't want the G doll, in how many different ways can he give the gifts?

You'll find my step-by-step solution here: http://www.beatthegmat.com/p-amp-c-problem-t269505.html

Cheers,
Brent