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

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One minus none is all.

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P(all complete) = 0.05

P(None complete)=1−P(All complete)
=1−0.05=0.95

If all the Penguins migrating has a probability of 0.05, none migrating has a probability 0.95. Because n is fixed and the event space only includes two outcomes: all succeed or all fail.

Thus, the correct answer is D. 0.95.

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30 Jun 2025, 19:39

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The answer is 0.95.

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

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Cannot be determined from the given information. We know only that p(n)=0.05. 1-0.05 would give at least some penguins do not migrate.

Bunuel

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30 Jun 2025, 19:58

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E. Cannot be determined from the given information

To answer the question we should know the number of penguins to determine the joint probability that none of the penguins complete the migration.

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For each individual penguin:
Success probability (completes migration): m
Failure probability (doesn't complete migration): 1-m

For a colony of p penguins:
All succeed: m × m × ... × m (p times) = m^p
All fail: (1-m) × (1-m) × ... × (1-m) (p times) = (1-m)^p

Knowing only that m^p=0.05 is insufficient to calculate (1-m)^p. We would need additional information about either m or p specifically.

Correct answer is E

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30 Jun 2025, 20:40

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We don't know the number of penguins included in the group (the all).

0.05 is the result of mutipliying each individual probability of completing the migration, let's call it p. So the correct math way to read is the following: p*p*p*p*.......*p =0.05. Where p repeats "n" times = the number of penguins in the group.
And for find the prob. that none complete migration, I must multiply the individual complement probability n times: (1-p)*(1-p)*(1-p)*(1-p)*.......*(1-p) =0.05. As I dont have the values of p and n, it is not possible to solve.

Also, the probability that none complete migration is not the complement because there are another scenarios to consider:
P(none) + P (just 1) + P (just 2) + P (just 3) + ....+ P (all) =1

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Let’s say each penguin has a probability ppp of completing the migration.
If there are nnn penguins, then:

The probability that all succeed = pn=0.05p^n = 0.05pn=0.05
The probability that none succeed = (1−p)n(1 - p)^n(1−p)n

Try p≈0.55, then p5≈0.05 → that fits.

Now plug into (1 - p)^5 → still ≈ 0.05

Answer:0.05, If all succeed with 5% chance, and each acts independently with equal chance, then the chance that none succeed is also 5%.

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Each penguin has a probability p of completing the migration, and there are n penguins. Then the probability that all of them complete it would be:
p^n = 0.05

If we negate All, we get Not all instead of None.

The probability that none of them make it would be: (1 - p)^n
Without knowing either p or n, we can't figure out the probability that none make it.

Therefore, I will go with option E.

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We’re told that each penguin has the same probability of completing the migration and that the probability of all penguins completing it is 0.05. If we let p represent the individual probability of a penguin completing migration and
n represent the number of penguins, then:
p^n=0.05

We are asked to find the probability that none of the penguins complete the migration. That would be:
(1−p)^n

At first glance, it might seem like we have enough information. But in reality, we only know the value of
p^n, and not p or n individually. There are infinitely many combinations of p and n
n that satisfy the equation and each combination would yield a different value for (1−p)^n
Even though the probability is equal for each penguin, without knowing either how many penguins there are or what the individual probability p is, we cannot compute the probability that none complete the migration.

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30 Jun 2025, 21:04

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Here, we know that the probability of all penguins completing the migration is 0.05. We can immediately infer that the probability of atleast 1 penguin NOT completing the migration is 0.95.

Since each penguin has the same individual probability of completing it, $p^n = 0.05 $
Thus the probability that none of the penguins complete the migration will be $(1 - p)^n$.

We cannot determine this from the information provided. Option E.

Bunuel

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Quote:

Let us assume that probability of completing the migration for each penguin is P, and there a total of N penguins.

Therefore, it's given that P^N = 0.05

Now, we have to find the probability that none of the penguins complete the migration.

Which means = probability of each penguin not completing the migration ^ number of penguins

= (1-P)^n

From the given info, this can't be determined.
Hence, the answer is (E)

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The likelihood that a single penguin successfully migrates is some unknown value. For multiple penguins, the chance that every one of them completes migration equals this individual success probability multiplied by itself repeatedly for each penguin, which we know equals 0.05.

Similarly, the probability that a given penguin fails to migrate is one minus that individual success probability. The chance that all penguins fail would be this individual failure probability multiplied by itself for each penguin.

Crucially, knowing only the combined success probability for all penguins does not allow us to determine the combined failure probability. These two group probabilities relate differently to the individual penguin's chances, and the all-fail scenario depends on information that isn't contained in the all-succeed probability alone. Many different scenarios could produce the same all-succeed probability while yielding completely different all-fail probabilities.

The right answer is E

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Answer: E
We cannot know because we do not have the number of penguins (n).
As we see further below, the answer varies according to n:

n = 1 penguin: 0.95
n = 2 penguins: 0.60
n = 3 penguins: 0.25

Some may fall in the trap answer D. Let's see why it is not the correct answer:

If p (all penguins will complete their annual migration) = 0.05, Then the opposite is 1 - (0.05) = 0.95.
But what exactly is the opposite of all? This is a typical negation question: NEG(all) = not all.
The trap here is to be misled by the qualifier none. Notice that the opposite of "all" is not none. But rather, none is a subset of not all.

So how do we go from p(not all) to p(none)? Well, it depends. Suppose we have:

1 penguin: then, p(none) coincides with p(not all) = 0.95, because there is only 1 penguin in total. If they can't complete, none can.

2 penguins: then, we also have to remember that

Quote:

If each penguin has the same probability of completing the migration

then p (1st penguin completing) = p (2nd penguin completing) = p(c). And also: $p(c)^2 = 0.05$.

What does that mean? That the probability of both completing it = p (all completing) = 0.05.

1. $p(c) = \sqrt{0.05} \approx{0.2236}$
2. $p(NEG c) = 1 - 0.2236 = 0.7764$
3. p(none) = $p(NEG c) * p(NEG c) = 0.7764^2 \approx{0.60}$

3 penguins: same process, but instead of $p(c)^2$, it is $p(c)^{3} = 0.05$.
1. $p(c) = \sqrt[3]{0.05}$ $\approx{0.3684}$. 2. $p(NEG c) = 1 - 0.3684 = 0.6316$. 3. p(none) = $p(NEG c) * p(NEG c) * p(NEG c)$ = $0.6316^{3}$ $\approx{0.25}$.

Hope to have made it clear.
Cheers,
DD

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The probability that all the penguins complete the migration is 0.05. If each penguin has the same probability of completing the migration, then let that probability be p. If there are n penguins, then p^n = 0.05. The probability that none complete the migration is (1 - p)^n. Since we don’t know the value of n, we cannot calculate the exact probability. So, the correct answer is E. Cannot be determined from the given information.

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how to join test and answer question?

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Bunuel

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Given:
P(All) = 0.05

To find:
P(None)

P(Not All) = 1 - P (All) = 0.95

But P (Not All) includes both P(None) & P(at least one) [P (Not All) = P (None) + P(At least 1)]

Since we don't know anything about number of penguins or P (at least 1), we cannot determine

Alternatively,

Let probability of 1 penguin completing migration be x; assuming the variable is independent. Let n be the number of penguins.

So, P (all) = $x^n$ = 0.05

x = $0.05^(1/n)$

P (none) = $(Probability of one penguin not completing migration)^n$ = $(1-x)^n$
= [m] (1-0.05^(1/n))^n [/m)

Since, we don't know n we cannot determine P (none)

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Bunuel

This question was provided by GMAT Club
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Let $p$ be the probability of each penguin completing their migration, and let $n$ be the total number of penguins in the group.
Now, Probability of all the penguins completing their migration will be $p*p*...(n times) = p^n = 0.05$ (given)

Now, Probability of none of the penguins complete the migration = $(1-p)(1-p)...(n times) = (1-p)^n$
Therefore, it is not possible to determine the probability unless we have either the value of p or n along with the given probability.

Correct answer is E. Cannot be determined from the given information

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Bunuel

This question was provided by GMAT Club
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Let p = probability that one penguin completes the migration
Let n = number of penguins in the group

We are told that the probability all n penguins succeed is
$ p^n = 0.05.$

The probability that none of the penguins succeed is
$(1 – p)^n.$

Because neither p nor n is given, (1 – p)^n has no single value.

Example checks
• If n = 1 ⇒ p = 0.05 ⇒ $(1 – p)^1 = 0.95 $
• If n = 2 ⇒ p = √0.05 ≈ 0.224 ⇒ $(1 – p)^2 ≈ 0.603 $

Different (p, n) pairs satisfy $p^n = 0.05$, but they give different results for $(1 – p)^n$.
Therefore the probability that none complete the migration cannot be determined from the information provided. The correct answer is E.

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Bunuel

This question was provided by GMAT Club
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Let
1. Probability of each penguin completing its annual migration = P
2. Number of penguins in the study = N

As per the question
P x P x P ........... x P = 0.5 (P is multiplied 'N' times as there are N penguins)
=> P^N = 0.5
=> P = 0.5^(1/N)

Now, we have to find out the probability that none of the penguins complete their annual migration.

Probability of a single penguin not completing its annual migration = (1 - P) = (1 - 0.5^(1/N))

Hence, Probability of all penguins not completing their annual migration = (1 - P)^N = (1 - 0.5^(1/N))^N

However, we need the number of penguins 'N' to be able to figure out the required probability. As the number of penguins increase the probability decreases.

IMHO Option E