A jar contains 6 Magenta balls, 3 Tan balls, 5 Gray balls

Question

A jar contains 6 Magenta balls, 3 Tan balls, 5 Gray balls and 7 Turquoise balls. Two balls are chosen from the jar. What is the probability that both balls chosen are Tan?

A. 1/70
B. 2/49
C. 1/21
D. 6/441
E. 1/49

Don't we have to assume that the balls are identical? If they are, then the number of ways of choosing 2 Tan identical balls out of 3 tan balls = 1. Isn't it?
If I assume that the balls are not withdrawn "at a time", OA is correct. However, how do I know whether the balls are withdrawn at a time or one by one?

EvaJager · Answer

The assumption is that balls of the same color are identical.
OA answer is correct anyway. It doesn't matter that you stick both your hands in a jar and remove two balls together or get them one-by-one.
What matters is the final result: the types/colors of the two chosen balls.

If you use probabilities: first ball Tan 3/21, second ball Tan 2/20, altogether (3/21)*(2/20) = 1/70.

Combinatorics: total number of ways to choose 2 balls out of 21 is 21C2 = 21*20/2 = 21*10. Total number of ways to choose 2 Tan balls out of 3 is 3C2 = 3*2/2 = 3.
Therefore, the required probability is 3/(21*10) = 1/70.

Answer A.

voodoochild · Answer

Ok. But what about this one:
a-bag-has-4-blue-3-yellow-and-2-green-balls-the-balls-of-139818.html

In this example, the number of ways of choosing any one out of 3 Yellow balls = 1+1+1+1 (no ball is chosen) instead of (3C0=1)+(3C1=3)+(3C2=3)+(3C3=1)=8 ways

Can you please explain the difference between these two questions?

voodoochild · Answer

Eva, Sorry to open this old thread. But, in this post : math-combinatorics-87345.html , the poster has mentioned that "Number of ways to pick 0 or more objects from n identical objects = n + 1" -- Hence, the number of ways to pick 2 balls out of say 5 identical balls must NOT be 5C2 -- Correct? Please help me. thanks

g81 · Answer

If it is not stated, we should assume without replacement,
Therefore: 3/21 x 2/20 = 1/70

Answer A

EvaJager · Answer

The wording  "Number of ways to pick 0 or more objects from n identical objects = n + 1"  is not clear.
It is meant that from n identical objects, we can choose none (which is 0), 1, 2,..., or all n objects - therefore we have n+1 choices.
For example, if we have 5 identical balls, we can choose 0, 1, 2, 3, 4, or 5 balls. We have a total of 5 + 1 choices.

Once we made up our mind how many balls we want to choose, say 2, then the number of possibilities to choose them is 5C2 = 5*4/2. 
Because, for the first ball 5 possibilities, for the second 4 possibilities, and then we have to divide by 2!, because the balls are identical, order doesn't matter (I would say it is meaningless, red is red, so why should I care which red I picked first?).

rrsnathan · Answer

Eva, Sorry to open this old thread. But, in this post : math-combinatorics-87345.html , the poster has mentioned that "Number of ways to pick 0 or more objects from n identical objects = n + 1" -- Hence, the number of ways to pick 2 balls out of say 5 identical balls must NOT be 5C2 -- Correct? Please help me. thanks The wording "Number of ways to pick 0 or more objects from n identical objects = n + 1" is not clear. It is meant that from n identical objects, we can choose none (which is 0), 1, 2,..., or all n objects - therefore we have n+1 choices. For example, if we have 5 identical balls, we can choose 0, 1, 2, 3, 4, or 5 balls. We have a total of 5 + 1 choices. Once we made up our mind how many balls we want to choose, say 2, then the number of possibilities to choose them is 5C2 = 5*4/2. Because, for the first ball 5 possibilities, for the second 4 possibilities, and then we have to divide by 2!, because the balls are identical, order doesn't matter (I would say it is meaningless, red is red, so why should I care which red I picked first?). Hi, i have a small doubt in this. This example of 5 balls and pick up 2 ball can be written as like this also right Probablity to take first ball is : 2/5 Probablity to take second ball: 1/4 so independent event = 2/5* 1/4= 1/10 which is reciprocal of wat u got. . is this correct answer.? i am trying to understand this in probablity ways/ pleas healp me Thanks, Rrsnathan

Narenn · Answer

Probability =  \frac{Desired Outcomes}{Total Outcomes}

Desired Outcomes = Number of ways of choosing 2 tan balls from 3 Tan balls = 3C2

Total Outcomes = Number of ways of choosing any 2 balls from 21 balls = 21C2

Thus Probability =  \frac{3C2}{21C2}  =  \frac{1}{70}

Asifpirlo · Answer

Both balls are tan = 3/21 × 2/20 = 1/70

bumpbot · Answer

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A jar contains 6 Magenta balls, 3 Tan balls, 5 Gray balls

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