GMAT Club Olympics 2025 (Day 1): The probability that all the penguins : Problem Solving (PS)

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Bunuel

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The answer to this question is option E.

Over here, the probability of all is given.
P(All)= 0.05.
Now, assume the probability of one penguin completing the migration is p.
The number of penguins is n.
Then p^n=0.05.

Now, for P(none).
The probability that one penguin doesn't complete migration is (1-p).
Then probability that none of them complete is (1-p)^n.

Here, we at least need the value of n to determine the value of p.
Since p=0.05^1/n

Hence, the probability cannot be determined.

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Since the number of penguin is not specified we cannot calculate the Prob for for each not completeing the task. Hence, we cant calculate the Prob that none complete the migration

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The probability that all the penguins in a marine study group complete their annual migration is 0.05. If each penguin has the same probability of completing the migration, what is the probability that none of the penguins complete the migration?

A. 0.05
B. 0.5
C. 0.75
D. 0.95
E. Cannot be determined from the given information
E should be the answer;
Without knowing the number of penguins , we cant determine the probability (none of the penguines complete the migration).

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30 Jun 2025, 10:37

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probability of (none of the penguins complete the migration) = 1 - 0.05 = 0.95

Quote:

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30 Jun 2025, 10:39

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Ans: E
Since we do not know how many penguins there are so we can not determine the P(none migrate)

A. 0.05
B. 0.5
C. 0.75
D. 0.95
E. Cannot be determined from the given information

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30 Jun 2025, 10:43

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The probability that all the penguins complete the migration is 0.05.

Also, each penguin has the same individual probability of completing it.

Let’s say that the probability is p for one penguin.
Now, imagine there are n penguins in total. Then:
Probability that all penguins complete the migration=p^n=0.05
That just means all penguins make it = p * p * p... n times.)
If that looks abstract, think of this like flipping a coin.

Say the probability of getting heads in one toss is 0.5.
Then the probability of getting heads 3 times in a row is:
0.5 * 0.5 * 0.5 = 0.5^3 = 0.125

Same logic here—each penguin has an independent chance p of making it, so all of them making it is p^n.
Now, the question is asking:
What’s the probability that none of the penguins complete the migration?
If one penguin’s chance of completing is p, then the chance it doesn’t complete is (1 - p).
So, the probability that none of the n penguins complete it is:
(1 - p)^n

Without knowing the number of penguins (n), we can’t find an exact numeric answer for the probability that none of them make it.
P.S. This is my first ever post on GMAT Club, still learning and finding my way around! 😊
Would love to hear any feedback or thoughts!

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30 Jun 2025, 10:52

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Total probability= probability of all penguins completing migration+ probability of none of the penguins completing migration

1= 0.05+ probability of none of the penguins completing migration
P(none of the penguins complete migration)= 0.95

Option D

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Let, there be n penguins. Each penguin has a probability p of completing migration.
Since they are independent, the probability that all penguins complete migration is:
P(all complete)=p^n=0.05
We want to find the probability that none of the penguins complete migration.
This means:
P(none complete)=(1 - p)^n
But we only know: p^n=0.05
We do not know the value of p or n individually, only their combination.
That means we cannot calculate (1 - p)^n directly from the given data.
Therefore answer is E.

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-Probability of one penguin completing immigration is: “p”

-number of penguins in the group: “n”

=>The probability of all penguins complete immigration: p^n= 0.05

-the probability that none of them complete the immigration: (1-p)^n

*As we don’t know the total number of penguins, we can’t calculate (1-p)^n

✅The correct answer is “E”

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30 Jun 2025, 11:08

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P(ALL penguins complete annual migration) = .05
Thus, 1 - P(ALL penguins complete annual migration)= P(No penguin completes the migration)

1 - .05 = .95
D

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Bunuel

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Given:

Probability that all penguins complete migration = 0.05.
Each penguin has same probability of completing migration, say p.
Let there be n penguins.

[hr]
Probability that all penguins complete migration:
p^n = 0.05.
[hr]
Probability that none of the penguins complete migration:
(1 - p)^n
[hr]
Can we find (1 - p)^n using only p^n = 0.05?
No — because without knowing either p or n, we cannot determine 1−p or (1 - p)^n.
We only have one equation and two unknowns ( p and n ).

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30 Jun 2025, 11:26

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probably incorrect, but my thought process is: Prob[ALL] = 1 - Prob[none]; therefore, .05= 1 - x and x = .95, but using outside knowledge it seems like it wouldn't make much sense for an annual migration to have a 95% fail rate lol but perhaps my fundamental understanding of probability is flawed.

So I'm inclined to choose E, but that seems like more of a trick, so I'll settle on D. 95%

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Given, we have n penguins in a study, and the probability that all the penguins complete the migration is P(all migrate) = .05. Additionally, each penguin has the same probability to migrate, P(Penguin 1 migrates) = P(Penguin 2 migrates) = ... = P(Penguin n migrates).

Asked to find the probability that none of the penguins migrate, P(none migrate).

P(Penguin 1 does not migrate) = 1 - P(Penguin 1 migrates) (treating this a Bernoulli trial).
Likewise, P(Penguin 1 does not migrate) = P(Penguin 2 does not migrate) = ... = P(Penguin n does not migrate) = 1 - P(Penguin 1 migrates).

P(none migrate) = (1 - P(Penguin 1 migrates))^n. We need to find n and P(Penguin 1 migrates). Unfortunately, from the given information:
P(all migrate) = (P(Penguin 1 migrates))^n = .05. We have one equation but two unknowns, and, therefore, more information is needed to solve this problem.

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30 Jun 2025, 11:39

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P(all) = 0.05 => P(all) = p^n = 0.05 => p = 0.05^{1/n}, where p == probability a penguin succeeds, and n == number of penguins.

The probability that none of the penguins complete the migration (q = 1-p):

P(none) = q^n = (1−p)^n

and p = 0.05^(1/n)

P(none)=(1−0.05^(1/n))^n

We cannot determine the exact probability, because we need n.

Answer: (E) Cannot be determined

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30 Jun 2025, 11:50

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P(none) = 1 - P(all) = 1 - 0.05 = 0.95

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30 Jun 2025, 11:55

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Hi! Alright, so I was a quick trigger here and made an obvious error:

At first, I thought it was as simple as taking the total probability (1) - the probability of all (0.05), thus finding out the probability of no penguins making it.
This however, is obviously wrong

I calculated the probability of not every single penguin making it, not that none of them would make it.

This probability is too difficult to say without knowing how many penguins there are. If we knew that, we could try calculating it with the binominal coefficient.

Correct answer: E

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30 Jun 2025, 12:34

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P event not happening = 1 - p event happening
= 1 - 0.05
= 0.95

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30 Jun 2025, 12:58

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Intuitively I initially thought Probability of all penguins migrate + Probability of no penguins migrate = 1 but that is wrong because some penguins can migrate so when we have P(All Migrate) = P(1 Migrate)^n where n is unknown so we know (E) is the correct answer.

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30 Jun 2025, 12:58

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According to the given question we can form (1/x)^y = 0.05, where x is the denominator of the rate and y the number of the penguins that are relevant for this question.

To give an answer, we can rewrite it to (1/x) = sqrt(y)(0.05).
But we are still missing the two relevant variables, as we are searching for (1/(1-x))^y.

Due to the given contraints, the questions is therefore not answerable.

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30 Jun 2025, 13:13

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According to my logic, the probability of all penguins completed their migration is 0.05 and none completed their migration will be 1 - 0.05 = . 95.

Also, verify with expert's solutions too.