A box contains 1 blue ball, 1 yellow ball,

Previous question was great and this one is even better. Thanks fo posting!

Approach #1 (Using compliment)

We'll find the probability of a compliment of our statement, that is 2nd ball is not yellow and 3d is yellow which is identical to the statement that 3d is yellow because we have only one yellow ball. We have 3 cases:

Blue Red Yellow + Red Blue Yellow + Red Red Yellow

\(\frac{1}{4}*\frac{2}{3}*\frac{1}{2} + \frac{2}{4}*\frac{1}{3}*\frac{1}{2} + \frac{2}{4}*\frac{1}{3}*\frac{1}{2} = \frac{3}{12} = \frac{1}{4}\)

Our sought-for probability will be \(1 - \frac{1}{4} = \frac{3}{4}\)

Answer C B

Approach #2 (Combinatorics)

"The second ball is yellow" means that we can't choose balls for the second draw, but we can choose balls for the first and the third draw. We need to chose 2 balls from remaining 3 (with exception of yellow) 3C2. Total number of cases will be 4C3.

\(\frac{3C2}{4C3} = \frac{3}{4}\)

Answer C. B

Probability that 2nd ball is yellow:
P of 1st ball not Y * P of 2nd ball Y
3/4*1/3= 1/4

P of 3rd ball Not Y:
=1- p of 3rd ball Y
=1- P of 1st ball not Y*p of 2nd ball Not Y* P of 3rd ball Y
=1- 3/4*2/3*1/2
=1-1/4
=/4

So 2nd ball is yellow or the 3rd ball is NOT yellow= 1/4+3/4= 1
What did I do wrong?
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HINT: P(A or B) = P(A) + P(B) - P(A and B)

Cheers,
Brent

Another solution is to list all of the possible outcomes for the 3 selections:
- RRY
- RRB
- RBR
- RYR
- RYB
- RBY
- BYR
- YBR
- BRY
- YRB
- BRR
- YRR

Of the 12 possible outcomes, 9 meet one or both of requirements.
So, P(2nd ball is yellow or the 3rd ball is NOT yellow) = 9/12 = 3/4 = B

First of all, it's useful to recognize that P(2nd ball is yellow or the 3rd ball is NOT yellow) is the SAME as P(1st ball is yellow or the 2nd ball is NOT yellow)

Let's apply the OR probability rule: P(A or B) = P(A) + P(B) - P(A and B)
So, P(1st ball is yellow or the 2nd ball is NOT yellow) = P(1st ball is yellow) + P(2nd ball is NOT yellow) - P(1st ball is yellow AND the 2nd ball is NOT yellow)

P(1st ball is yellow) = 1/4

P(2nd ball is NOT yellow) = P(1st ball is NOT yellow) = 3/4

P(1st ball is yellow AND the 2nd ball is NOT yellow) = P(1st ball is yellow) x P(the 2nd ball is NOT yellow)
= 1/4 x 1
= 1/4

So, P(1st ball is yellow or the 2nd ball is NOT yellow) = P(1st ball is yellow) + P(2nd ball is NOT yellow) - P(1st ball is yellow AND the 2nd ball is NOT yellow)
= 1/4 + 3/4 - 1/4
= 3/4
= B

Cheers,
Brent

First of all, it's useful to recognize that P(2nd ball is yellow or the 3rd ball is NOT yellow) is the SAME as P(1st ball is yellow or the 2nd ball is NOT yellow)

Hi Brent, can you please help me understand highlighted statement. Thanks in advance!

Brent, please explain how you come up with P(the 2nd ball is NOT yellow)=1 in that step? as earlier it was =3/4..... it is confusing....

I understand this solution. But Is this only applicable in overlapping situation/s?
Sorry if I am talking foolish. P&C and probability.... uff!

Thanks very much, Brent.
_________________

Hi

You may have been asked two different things -
(a) 2nd is yellow or (b)3rd is not yellow-,
You should recognise that (a) is part of (b) so just find (b)..

3rd can be..
1) blue
The rest three are 2 red and 1 yellow so ways =3!/2!=3..
2) red
The rest three are 1 red, 1 blue and 1 yellow so ways =3!=6..
Total ways=3+6=9..

Overall ways=4!/2!=12..

Probability=9/12=3/4

B
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GMATPrepNow, Could you briefly explain why P(the 2nd ball is NOT yellow)=1?

P(2nd ball is NOT yellow) = P(1st ball is NOT yellow) = 3/4 , the question mention about WITHOUT REPLACEMENT. I am confused about this statement.
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Good question.
For AND probabilities, the formula is P(A and B) = P(A) x P(B|A), where "P(B|A)" stands for "the probability that event A occurs GIVEN THAT event B occurs"
So, P(1st ball is yellow AND the 2nd ball is NOT yellow) = P(1st ball is yellow) x P(2nd ball is NOT yellow given that the 1st ball is yellow)
So, if the 1st ball is yellow, then the probability is 1 that the 2nd ball is NOT yellow (since there's only 1 yellow ball).

Does that help?

Cheers,
Brent

\(1B\)
\(1Y\)
\(2R\)

the questions asks ==> Prob (2nd yellow) OR Prob (1st yellow ,or 2nd yellow, or no yellow at all)

Notice that the prob of 2nd yellow is counted twice, which means we can remove one of them. which leave us with

\(Prob (1st yellow + 2nd yellow+ No yellow)=1-(3rd yellow)\)

\((Y)*(RorB)*(RorB)+(RorB)*(Y)*(RorB)+(RorB)*(RorB)*(RorB)=1/4*3/3*2/2+3/4*1/3*2/2+3/4*2/3*1/2=3/4\)

same thing as:

\(1-(3rd yellow)=1-[(RorB)*(RorB)*(Y)]=1-(3/4*2/3*1/2)=1-1/4=3/4\)

Great Question.

One way to make both the 'listing all the options' and the 'sequence of events' approaches easier is to just think of the balls as belonging to only two types -- Yellow (Y) and Not Yellow (N).

Listing All The Options

If we include a fourth slot to represent the ball not chosen, I think it's easier to see that there are only four (equally likely) outcomes:

YNN N
NYN N
NNY N
NNN Y

Each of these sequences is equally likely (there's nothing that suggests that the yellow is any more likely to show up in any particular position), so we can just notice that in three of the options (1, 2, and 4 above) the yellow is either second OR not third. So there is a 3/4 chance we'll get one of the three permissible options.


Sequence of Events - Direct

There are only three sequences that have either the second ball yellow or have the third ball as not-yellow:

NYN
YNN
NNN

Plugging in numbers, we get:

NYN ----> (3/4)(1/3)(2/2) = 1/4
YNN ----> (1/4)(3/3)(2/2) = 1/4
NNN ----> (3/4)(2/3)(1/2) = 1/4

So the probability of getting one OR the other OR the other sequence = 1/4 + 1/4 + 1/4 = 1/2


Sequence of Events -- Indirect

There is only one way to get NEITHER of the scenarios we want. That's when the third ball IS yellow (otherwise it will definitely be the case that the third ball is not yellow (or the second is)):

NNY ---> (3/4)(2/3)(1/2) = 1/4

So the probability of NOT getting what we want is 1/4. Therefore the probability of getting what we want is 1 - 1/4 = 3/4

We are given that a box contains 1 blue ball, 1 yellow ball, and 2 red balls. We need to determine, when selecting 3 balls from the box, the probability that the second ball is yellow OR the third ball is not yellow.

Let’s look at the requirements in two separate scenarios:

Scenario 1: The second ball is yellow

The probability of not choosing the yellow ball in the first draw is 3/4, and the probability of choosing the yellow ball from the remaining 3 balls in the second draw is 1/3 (we do not care which ball is selected in the third draw in this scenario). Therefore, the probability of choosing the yellow ball in the second draw is:

3/4 x 1/3 = 1/4

Scenario 2: The third ball is not yellow. There are several ways this can happen, as follows:

a. The first ball is yellow. If this happens, we know that the third ball cannot be yellow because there is only one yellow ball in the box.

Since there are four balls, the probability of choosing the yellow ball in the first draw is 1/4 (we do not care which balls are selected in the second or third draws in this scenario).

b. The second ball is yellow. If this happens, we know that the third ball cannot be yellow because there is only one yellow ball in the box. We already considered this outcome in Scenario 1, so we WILL NOT include it again here.

c. The yellow ball is not selected at all. This outcome also satisfies the requirement that the third ball is not yellow.

The probability of choosing a non-yellow ball in the first, second, and third draws is 3/4, 2/3, and 1/2, respectively. So the probability of not choosing a yellow ball in three draws is:

3/4 x 2/3 x 1/2 = 1/4

Since any of these outcomes would satisfy the given condition and they are mutually exclusive, we can sum the individual probabilities of each outcome:

1/4 + 1/4 + 1/4 = 3/4

Answer: B
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P(2nd yellow) = first not yellow * second yellow = (3/4) * (1/3) = 1/4

P(3rd not yellow) = first yellow + second yellow + no yellow at all
= 1/4 + 1/4 + (3/4)* (2/3) * (1/2) = 3/4

P(2nd yellow AND 3rd not yellow) = P(choosing yellow in second draw) = 1/4

P(2nd yellow OR 3rd not yellow) = P(2nd yellow) + P(3rd not yellow) - P(2nd yellow AND 3rd not yellow)
= (1/4) + (3/4) - (1/4) = 3/4

This is how I solved it:


We need to get the probability of 1) 2nd is yellow, 2) 3rd is not yellow

1)2nd is yellow:
(3/4)*(1/3) = 1/4

2) 3rd is not yellow = 1st is yellow + 2nd is yellow + 4th is yellow
However, since '2nd is yellow' is already counted in the first step, we only need to get 1st is yellow+4th is yellow.
1st is yellow: 1/4
4th is yellow: (3/4)*(2/3)*(1/2)=1/4

The answer should be the answer from the 1st step + 2nd step = 1/4 + 1/4 + 1/4 = 3/4


This is basically the long way of saying "1 - probability of 3rd is yellow", but I found this way easier to understand.

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