<Question>
A pirate searches seven islands for buried treasure. If each island has a $\frac{1}{5}$ chance of having treasure, what is the probability that exactly 4 of the islands have treasure?
</Question>

<think>
<Plan_1>
Okay, so I need to figure out the probability that exactly 4 out of 7 islands have buried treasure, given that each island has a 1/5 chance of having treasure. Hmm, let's start by recalling what kind of probability problem this is. 

First, each island is an independent event, right? The chance of treasure on one island doesn't affect the others. Each has a probability of 1/5, and the complement, not having treasure, would be 4/5. Since we're looking for exactly 4 successes (treasure) out of 7 trials (islands), this sounds like a binomial probability problem. The binomial formula gives the probability of having exactly k successes in n trials, which is:

<prompt_1.1> Recall the binomial probability formula and identify the parameters for this problem. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
P(k) = C(n, k) * p^k * (1-p)^(n-k)

Where C(n, k) is the combination of n things taken k at a time. So in this case, n is 7, k is 4, p is 1/5.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Let me write that out. First, calculate the combination C(7, 4). That's the number of ways to choose 4 islands out of 7 to have treasure. The formula for combinations is:

<prompt_2.1> Calculate the combination C(7, 4) using the combinations formula. </prompt_2.1>

Next, we need to calculate p^k, which is (1/5)^4. Let me compute that.
(1/5)^4 is 1^4 / 5^4 = 1 / 625.

Then, (1-p)^(n-k) is (4/5)^(7-4) = (4/5)^3. Let me calculate that as well.

<prompt_2.2> Calculate (1/5)^4 and (4/5)^3. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
C(7, 4) = 7! / (4! * (7-4)! ) 

So 7 factorial divided by (4 factorial times 3 factorial). Let me compute that. 

7! is 7*6*5*4*3*2*1 = 5040. But since 4! is 24 and 3! is 6, so 24*6 = 144. Then 5040 divided by 144. Let me do that division: 5040 ÷ 144. Hmm, 144*35 = 5040, right? Because 144*30 = 4320, and 144*5 = 720, so 4320+720=5040. So 35. So C(7,4) is 35.
</execution_2.1>

<execution_2.2>
4^3 is 64, and 5^3 is 125, so that's 64 / 125.

Now, putting it all together: C(7,4) * (1/5)^4 * (4/5)^3.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1, execution_2.2:
So that's 35 * (1/625) * (64/125). Let me compute this step by step. 

First, multiply 35 and 1/625.

<prompt_3.1> Multiply 35 and 1/625, then multiply the result by 64/125. </prompt_3.1>
-----

Alternatively, maybe I can check using decimal equivalents to verify. Let's see:

<prompt_3.2> Verify the result using decimal approximations. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
35 divided by 625. Let me see, 35/625. Hmm, 35 divided by 25 is 1.4, so 35 divided by 625 is 0.056. Wait, 625*0.056 = 35, yes. So 0.056. Then multiply that by 64/125. 

Wait, 64/125 is 0.512. So 0.056 * 0.512. Let me compute that. 0.056 * 0.5 is 0.028, and 0.056 * 0.012 is 0.000672. Adding those together: 0.028 + 0.000672 = 0.028672. So approximately 0.028672. To convert that back to a fraction, but maybe it's better to compute it exactly.
</execution_3.1>

<execution_3.2>
(1/5) is 0.2, (4/5) is 0.8. So (0.2)^4 = 0.0016, (0.8)^3 = 0.512. Then 35 * 0.0016 * 0.512. 

First, 35 * 0.0016 = 0.056. Then 0.056 * 0.512. Let's compute that. 0.056 * 0.5 = 0.028, and 0.056 * 0.012 = 0.000672. Adding those gives 0.028 + 0.000672 = 0.028672. 

Then converting that back to a fraction. 0.028672. Let's see, 0.028672 * 100000 = 2867.2, so maybe 28672/1000000. But this might not be helpful. Alternatively, since we already did the exact fraction calculation and got 3360/78125, which is 0.028672, that's the same as the decimal.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Let me do it fractionally. 

35 * (1/625) * (64/125) = 35 * 64 / (625 * 125). Let's compute numerator and denominator.

<prompt_4.1> Calculate the exact probability by performing the multiplication fractionally. </prompt_4.1>
-----

Based on execution_3.2:
So the probability is 3360/78125, which simplifies to 672/15625, which is approximately 0.028672 or 2.8672%.
Therefore, the exact answer is 672/15625, which is 3360/78125. Either form is correct, but maybe 672/15625 is preferable as it's reduced.
</Plan_4>

<Execution_4>
<execution_4.1>
35*64 = 2240. 625*125 = 78125. So the probability is 2240 / 78125. Let me check if this fraction can be simplified. 

2240 and 78125. Let's see, 2240 divides by 5: 2240 ÷ 5 = 448. 78125 ÷ 5 = 15625. So 448/15625. Let's check if 448 and 15625 have any common factors. 15625 is 5^6, since 5^6 = 15625. 448 is 64*7, which is 2^6 *7. So no common factors with 5. Therefore, 448/15625 is the simplified fraction. 

So the exact probability is 448/15625. Let me check that 35*64=2240, and 625*125=78125, then divided by 5 gives 448/15625.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
But let me confirm the steps again to ensure I didn't make a mistake. 

First, combination C(7,4)=35. Correct. Then (1/5)^4=1/625.

<prompt_5.1> Re-verify each step of the calculation to ensure accuracy. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
Correct. (4/5)^3=64/125. Correct. Then multiply 35*(1/625)*(64/125). Yes, 35*64=2240, and 625*125=78125. Then 2240/78125 divided by 5 is 448/15625. 

Alternatively, if I leave it as 2240/78125, that's also a correct answer, but it's better to simplify it. Since 2240 ÷5=448 and 78125 ÷5=15625, and 448 and 15625 have no common factors, that's the simplified form.
</execution_5.1>
</Execution_5>

<Final_answer>
Therefore, the probability that exactly 4 of the islands have treasure is $\boxed{\frac{3360}{78125}}$.
</Final_answer>
</think>