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

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We are missing one information: the number of penguin, to determine the probability for one penguin, since we know it is the same for all of them.

All the penguins complete the migration: P= 0.05, Not All: P=0.95 (could be ALL - 1, or -10000)

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Bunuel

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

Let's assume that there are N penguins and each have P probability to complete migration.
Then 0.05 = (P)^N
what we need is (1-P)^N
This gives us the probability that none of the penguins complete migration.
We cannot compute this without the unknown values.
E

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Without knowing the number of penguins, we cant determine the probability of all penguins failing.

The trap answer here is 0.95 but that give us the probability of at least one penguin failing (P at least 1 failing) = 1 - P (all completing).

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The Probability of all penguins = 0.05

All penguins mean there is more than 1 so, let the number of penguins be n

Now, p(none of the penguins) = 1-P(All penguins)^n(Power n) =(1-0.05)^n(Power n)

We don't know n. Hence can't be determined

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Probability of each penguin complete their migration is p, then all complete their migration is
$p^n = 0.05.$
Probability that each penguin do not complete the migration is (1-p)

none of the penguins complete the migration is$ (1-p)^n $

Cannot determine 2 variables with 1 equation given.

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30 Jun 2025, 10:01

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D. as the probability of event occurring is 0.05 probability of event not occurring is 1-0.05=0.95

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This is an easy question, but the fact that we have fallen for traps before makes one hesitant to mark the answer right away for these types of questions.
Coming to the question, it is given that the probability that all (n) penguins complete their annual migration (cam) = 0.05
Let the Probability that 1 penguin cam = p
Then the Probability that n penguins cam = p^n

Therefore, p^n = 0.05

Now the probability that 1 penguin cannot cam = 1-p
probability that n penguins cannot cam = (1-p)^n

we cannot determine both n & p, hence insufficient information
Answer - E

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let n penguins with each having p probability of success

given: p^n =0.05

reqd: (1-p)^n

cannot be calculated as details of those group who have succeed & failed not given.

Ans E

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Approach :
Assuming 'n' to be the number of penguins; we know P(n) = 0.05 ;probability of completing the migration
P(none) + P(1) + P(2) + ..... P(n) = 1
P(none) = 1-0.05 - P(1) - P(2) -.....
even though each penguin has same probability of completing migration we cannot determine that Probability(P) or number of penguins 'n' from P(n) = n*P
hence insufficient

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Bunuel

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each penguin’s success rate be p, and there are n penguins.
Then:
p^n=0.05 (all complete)
Probability none complete = (1 - p)^n
But without knowing p or n, probability of none of the peguines complete migration cannot be calculated.
So, answer is (E) Cannot be determined

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Probability that none would complete the migration =1-probability of all completing migration
So, 1-0.05=0.95

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The probability that none of the penguins complete the migration can't be determined from the information given, since the complement of it is the probability of at least one penguin doesn't complete the migration, which isn't provided in the question. Each scenario must equal to 1 (all penguin complete the migration + at least one penguin doesn't complete the migration). Thus, the correct answer is E.

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ANS E

The probability of all the penguins to complete the migrations is the product of all the penguins, as they have the same probability the formula will be:
probability of a penguin to complete the migration = (P)
so (P)*(P)*(P)*(P)*(P)*(P)* (P)*(P)..............(P)=.05 = (P)^n=.05
where "n" is the total number of penguins in the study.
The question ask for the probability the none of the complete de migration so the formula will be: (1-P)^n= ?
because we don't have "n" it cannot be determine.

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Probability of A = 1 - Probability of contrary of A
thus P of none of the penguins = 1 - P all of the penguins = 0.95

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We would require No. of penguins to calculate the probability of none of the penguins completing migration.

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Let the number of penguins in the study group be "n".
Since, the probability of each penguin completing the migration is same, considering the probability of each penguin completing the migration as "p". So the probability of each penguin not completing the migration is "1-p".

Given: Probability of all the penguins completing the migration is 0.05. Since, we are considering there are n penguins, this can be represented as p*p*p*....n times. So (p^n) = 0.05.

To Find: Probability of no penguin completing the migration. This can be represented as (1-p)*(1-p)*(1-p)....n times i.e. (1-p)^n

Now, we have 2 variables (p and n) and only one equation (p^n = 0.05). Additionally, for multiple +ve integral values of n, we get different possible values of p. Hence, imo, we cannot find the value of (1-p)^n.

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All succeed and None succeed are just two out of many possible outcomes, for example, there could also be exactly 1 succeeds.

Therefore, P(none succeed) ≠ 1 - P(all succeed).

IMO, E is the correct ans

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The probability that ALL the penguins in a marine study group complete their annual migration is 0.05, therefore NONE is 1-0.05 = 0.95.

Individual probabilities here are irrelevant, as we already have the probability of all.

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Can't determine as we don't know, how many penguins are there.

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Prob. (all penguin complete migration) = .05
Prob.( none complete) = P (Penguin 1 not completing migration) x P(Penguin 2 not completing migration) ....X P(Penguin n not completing migration)

even if we know that prob. of completing migration is same for all penguin we don't know how many penguins are there hence we cannot find individual penguin's migration probability. So we cannot find individual penguin's probability of not migrating and it follows that we cannot find prob(all penguin not migrating)