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Bunuel
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A box contains 10 balls, which are either red or white in color. 06 Jan 2025, 01:30
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A box contains 10 balls, which are either red or white in color. If two balls are selected at random from the box, what is the probability that the both the selected balls are white?

(1) The probability that one red ball and one white ball is selected is 16/45
(2) The probability that both the selected balls are red is 28/45

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A box contains 10 balls, which are either red or white in color. Updated on: 09 Jan 2025, 08:18
Originally posted by hr1212 on 06 Jan 2025, 05:48.
Last edited by hr1212 on 09 Jan 2025, 08:18, edited 1 time in total.
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Bunuel
A box contains 10 balls, which are either red or white in color. If two balls are selected at random from the box, what is the probability that the both the selected balls are white?

(1) The probability that one red ball and one white ball is selected is 16/45
(2) The probability that both the selected balls are red is 28/45

­
Show more
r + w = 10, calculate wC2/10C2 = wC2/45?

Statement 1 - rC1*wC1/45 = 16/45
r*w = 16
r*(10-r) = 16
r^2 - 10r + 16 = 0
(r-8)*(r-2) = 0
r = 8 or 2

which means no. of white balls are either 2 or 8. As we cannot get an accurate answer this statement is insufficient.

Statement 2 - rC2/45 = 28/45
(r*(r-1))/(2*45) = 28/45
r*(r-1) = 56
r^2 - r - 56 = 0
(r-8)*(r+7) = 0
r = 8 as no. of balls should be positive

which means no. of white balls are 2 and we should be able to effectively calculate value for wC2/45. Hence, this statement is sufficient.

IMO: B
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A box contains 10 balls, which are either red or white in color. 06 Jan 2025, 15:24
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Statement 1: honestly, I didn't use math for this which may mean I'm incorrect but if my logic is correct, this is what I was thinking. given the information that there is a 16/45 chance to get both a red and white ball, we have no information on if the remaining 29/45 chance is more heavily weighted towards red or white. therefore, since we cant determine if there is a higher chance of getting one ball or the other, Statement 1 is insufficient

Statement 2: you're able to find out what the chances are for getting a red ball if you were to decide to solve for it. since you can do that, then you'd also be able to find out what the chances are to getting both white balls. Statement 2 is sufficient

the answer is B
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A box contains 10 balls, which are either red or white in color. 07 Jan 2025, 00:36
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ccarson
Statement 1: honestly, I didn't use math for this which may mean I'm incorrect but if my logic is correct, this is what I was thinking. given the information that there is a 16/45 chance to get both a red and white ball, we have no information on if the remaining 29/45 chance is more heavily weighted towards red or white. therefore, since we cant determine if there is a higher chance of getting one ball or the other, Statement 1 is insufficient

Statement 2: you're able to find out what the chances are for getting a red ball if you were to decide to solve for it. since you can do that, then you'd also be able to find out what the chances are to getting both white balls. Statement 2 is sufficient

the answer is B
Show more
I would suggest that it's always safer to do math for DS questions unless you are short on time, because what I have observed for high quality questions is that they often lead to the default assumption trap where you assume something obvious, let's say a quadratic equation will have 2 roots and hence 2 solutions but on solving you might realize that it was a perfect square with a single root. Similarly, you would sometimes see that both the statements are providing same information if rearranged with some basic match but it ends up giving a mirage like feel and forcing people to choose obvious incorrect choice.

To clarify, let's tweak the probability value in statement 1 slightly. Let the probability to select one red and one white ball be equal to 25/45. Then by your logic, the remaining probability 20/45 also wouldn't give us any indication on which side the probability is more weighted on, and hence would seem insufficient. But now if we solve it mathematically, we get,

\(r*(10 - r) = 25\)
\(r^2 - 10r + 25 = 0\)
\((r - 5)^2 = 0\)
\(r = 5\)

which means no. of white balls are also 5, hence the statement would have been sufficient.
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A box contains 10 balls, which are either red or white in color. 09 Jan 2025, 07:05
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Could you explain the solution a bit more please? For both the statements, what I am unable to understand is how you are linking the numerators -

For the second statement - how did the RHS become 56?

Thanks in advance!
hr1212
Bunuel
A box contains 10 balls, which are either red or white in color. If two balls are selected at random from the box, what is the probability that the both the selected balls are white?

(1) The probability that one red ball and one white ball is selected is 16/45
(2) The probability that both the selected balls are red is 28/45

­
r + w = 10, calculate wC2/10C2 = wC2/45?

Statement 1 - r*w/45 = 16/45
r*w = 16
r*(10-r) = 16
r^2 - 10r + 16 = 0
(r-8)*(r-2) = 0
r = 8 or 2

which means no. of white balls are either 2 or 8. As we cannot get an accurate answer this statement is insufficient.

Statement 2 - rC2/45 = 28/45
r*(r-1) = 56
r^2 - r - 56 = 0
(r-8)*(r+7) = 0
r = 8 as no. of balls should be positive

which means no. of white balls are 2 and we should be able to effectively calculate value for wC2/45. Hence, this statement is sufficient.

IMO: B
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A box contains 10 balls, which are either red or white in color. 09 Jan 2025, 07:05
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Could you explain the solution a bit more please? For both the statements, what I am unable to understand is how you are linking the numerators - I understand the first line where you can calculating 10C2 and how that is 45, but how can we say r*w is 16. The numerator of 16/45, i.e., 16 is actually RC1 * WC1, is this how we are arriving at r*w =16?

For the second statement - how did the RHS become 56?

Thanks in advance!
hr1212
Bunuel
A box contains 10 balls, which are either red or white in color. If two balls are selected at random from the box, what is the probability that the both the selected balls are white?

(1) The probability that one red ball and one white ball is selected is 16/45
(2) The probability that both the selected balls are red is 28/45

­
r + w = 10, calculate wC2/10C2 = wC2/45?

Statement 1 - r*w/45 = 16/45
r*w = 16
r*(10-r) = 16
r^2 - 10r + 16 = 0
(r-8)*(r-2) = 0
r = 8 or 2

which means no. of white balls are either 2 or 8. As we cannot get an accurate answer this statement is insufficient.

Statement 2 - rC2/45 = 28/45
r*(r-1) = 56
r^2 - r - 56 = 0
(r-8)*(r+7) = 0
r = 8 as no. of balls should be positive

which means no. of white balls are 2 and we should be able to effectively calculate value for wC2/45. Hence, this statement is sufficient.

IMO: B
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A box contains 10 balls, which are either red or white in color. 09 Jan 2025, 08:21
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Yes, your understanding is correct.

Also added the missing step, hope that clarifies your query.
JohnAnjana
Could you explain the solution a bit more please? For both the statements, what I am unable to understand is how you are linking the numerators - I understand the first line where you can calculating 10C2 and how that is 45, but how can we say r*w is 16. The numerator of 16/45, i.e., 16 is actually RC1 * WC1, is this how we are arriving at r*w =16?

For the second statement - how did the RHS become 56?

Thanks in advance!
hr1212
Bunuel
A box contains 10 balls, which are either red or white in color. If two balls are selected at random from the box, what is the probability that the both the selected balls are white?

(1) The probability that one red ball and one white ball is selected is 16/45
(2) The probability that both the selected balls are red is 28/45

­
r + w = 10, calculate wC2/10C2 = wC2/45?

Statement 1 - r*w/45 = 16/45
r*w = 16
r*(10-r) = 16
r^2 - 10r + 16 = 0
(r-8)*(r-2) = 0
r = 8 or 2

which means no. of white balls are either 2 or 8. As we cannot get an accurate answer this statement is insufficient.

Statement 2 - rC2/45 = 28/45
r*(r-1) = 56
r^2 - r - 56 = 0
(r-8)*(r+7) = 0
r = 8 as no. of balls should be positive

which means no. of white balls are 2 and we should be able to effectively calculate value for wC2/45. Hence, this statement is sufficient.

IMO: B
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A box contains 10 balls, which are either red or white in color. 11 Jan 2025, 08:14
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Thanks! Saw the additional step, I understand the second step now. However, in the first statement also, should it not be 2*45 in the denominator? Thanks in advance!
hr1212
Yes, your understanding is correct.

Also added the missing step, hope that clarifies your query.
JohnAnjana
Could you explain the solution a bit more please? For both the statements, what I am unable to understand is how you are linking the numerators - I understand the first line where you can calculating 10C2 and how that is 45, but how can we say r*w is 16. The numerator of 16/45, i.e., 16 is actually RC1 * WC1, is this how we are arriving at r*w =16?

For the second statement - how did the RHS become 56?

Thanks in advance!
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A box contains 10 balls, which are either red or white in color. 11 Jan 2025, 13:30
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Bunuel
A box contains 10 balls, which are either red or white in color. If two balls are selected at random from the box, what is the probability that the both the selected balls are white?

(1) The probability that one red ball and one white ball is selected is 16/45
(2) The probability that both the selected balls are red is 28/45

­
Show more

For r red and w white: r + w = 10

S1: rC1 * wC1 / 10C2 = 16/45 => r*w = 16 => possible values are r = 8 and w = 2 or vice-versa - Insufficient
S2: rC2/10C2 = 28/45 => rC2 = 28 => r = 8 => w = 2 - Sufficient
Ans B
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A box contains 10 balls, which are either red or white in color. 11 Jan 2025, 13:45
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First statement has => \(\frac{rC1*wC1}{45} = \frac{r*w}{45}\), whereas,
Second statement has => \(\frac{rC2}{45} = \frac{r*(r-1)/2}{45} = \frac{r*(r-1)}{2*45}\)
JohnAnjana
Thanks! Saw the additional step, I understand the second step now. However, in the first statement also, should it not be 2*45 in the denominator? Thanks in advance!
hr1212
Yes, your understanding is correct.

Also added the missing step, hope that clarifies your query.
JohnAnjana
Could you explain the solution a bit more please? For both the statements, what I am unable to understand is how you are linking the numerators - I understand the first line where you can calculating 10C2 and how that is 45, but how can we say r*w is 16. The numerator of 16/45, i.e., 16 is actually RC1 * WC1, is this how we are arriving at r*w =16?

For the second statement - how did the RHS become 56?

Thanks in advance!
Show more
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A box contains 10 balls, which are either red or white in color. 13 Jan 2025, 13:13
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Thats semi-correct (when talking about the missing efficiency of a non-hard-math approach in this case).

Theres no order mentioned, in which the balls were grabbed. So we have two ways (W+B and B+W).
Without much further math, we can argue, that if the probability of 1B + 1W is 50/90, it must result in: (5/10*5/9+5/10*5/9).
That leads us to the conclusion, that both colours must be present in a quantity of five.

In general i agree with you, that a math-approach might be the safer way.
Thats my first day doing DS, but im certain that a mix is the way to go.

hr1212
ccarson
Statement 1: honestly, I didn't use math for this which may mean I'm incorrect but if my logic is correct, this is what I was thinking. given the information that there is a 16/45 chance to get both a red and white ball, we have no information on if the remaining 29/45 chance is more heavily weighted towards red or white. therefore, since we cant determine if there is a higher chance of getting one ball or the other, Statement 1 is insufficient

Statement 2: you're able to find out what the chances are for getting a red ball if you were to decide to solve for it. since you can do that, then you'd also be able to find out what the chances are to getting both white balls. Statement 2 is sufficient

the answer is B
I would suggest that it's always safer to do math for DS questions unless you are short on time, because what I have observed for high quality questions is that they often lead to the default assumption trap where you assume something obvious, let's say a quadratic equation will have 2 roots and hence 2 solutions but on solving you might realize that it was a perfect square with a single root. Similarly, you would sometimes see that both the statements are providing same information if rearranged with some basic match but it ends up giving a mirage like feel and forcing people to choose obvious incorrect choice.

To clarify, let's tweak the probability value in statement 1 slightly. Let the probability to select one red and one white ball be equal to 25/45. Then by your logic, the remaining probability 20/45 also wouldn't give us any indication on which side the probability is more weighted on, and hence would seem insufficient. But now if we solve it mathematically, we get,

\(r*(10 - r) = 25\)
\(r^2 - 10r + 25 = 0\)
\((r - 5)^2 = 0\)
\(r = 5\)

which means no. of white balls are also 5, hence the statement would have been sufficient.
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A box contains 10 balls, which are either red or white in color. 05 Mar 2026, 07:57
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