select the correct answer.
which statement is an inverse of p?
p: if two out of five people use the subway every day, then the probability of a person using the subway is 0.4.
a. if the probability of a person using the subway is 0.4, then two out of five people use the subway every day.
b. if two out of five people dont use the subway every day, then the probability of a person using the subway isnt 0.4.
c. if the probability of a person not using the subway is 0.4, then two out of five people dont use the subway every day.
d. if the probability of a person not using the subway is 0.4, then three out of five people use the subway every day.
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# Brief Explanations:
First, understand the original statement $ p $: "If two out of five people use the subway every day, then the probability of a person using the subway is 0.4". The inverse of a conditional statement "If $ A $, then $ B $" is "If not $ B $, then not $ A $". Here, $ A $: "two out of five people use the subway every day", $ B $: "probability of a person using the subway is 0.4". So the inverse should be "If the probability of a person using the subway is not 0.4, then two out of five people don't use the subway every day"? Wait, no, let's re - analyze. Wait, the original $ p $: Let's parse $ p $ correctly. $ p $: Hypothesis $ H $: two out of five use subway daily, Conclusion $ C $: probability of a person using subway is 0.4 (since $ 2/5 = 0.4 $). The inverse of "If $ H $, then $ C $" is "If not $ C $, then not $ H $". So "If the probability of a person using the subway is not 0.4 (i.e., probability of not using is 0.4? Wait, no, probability of using not 0.4. Wait, option B: "If two out of five people don't use the subway every day, then the probability of a person using the subway isn't 0.4" – no, that's not inverse. Wait, option C: "If the probability of a person not using the subway is 0.4, then two out of five people don't use the subway every day" – since probability of not using is 0.4, then probability of using is $ 1 - 0.4=0.6 $, so two out of five not using means three out of five using. Wait, maybe I made a mistake. Wait, original $ p $: "If two out of five use (so proportion $ 2/5 = 0.4 $), then probability of a person using is 0.4". The inverse is "If the probability of a person using the subway is not 0.4, then two out of five people do not use the subway every day". But looking at the options, option B: "If two out of five people don't use the subway every day, then the probability of a person using the subway isn't 0.4" – that's the contrapositive? No, contrapositive is "If not $ C $, then not $ H $", inverse is "If not $ H $, then not $ C $"? Wait, no, the definition of inverse of a conditional statement "If $ P $, then $ Q $" is "If $
eg P $, then $
eg Q $". So $ P $: two out of five use subway daily, $ Q $: probability of a person using subway is 0.4. So inverse is "If two out of five people do not use the subway daily ($
eg P $), then the probability of a person using the subway is not 0.4 ($
eg Q $)", which is option B? Wait, no, option B says "If two out of five people don't use the subway every day, then the probability of a person using the subway isn't 0.4". Let's check the options again:

Option A: "If the probability of a person using the subway is 0.4, then two out of five people use the subway every day" – this is the converse, not inverse.

Option B: "If two out of five people don't use the subway every day, then the probability of a person using the subway isn't 0.4" – Let's see, $
eg P $: two out of five don't use (so three out of five use, proportion $ 3/5 = 0.6 $), so $
eg Q $: probability of using isn't 0.4. So this is "If $
eg P $, then $
eg Q $", which is the inverse of "If $ P $, then $ Q $".

Wait, maybe I messed up the definition. The inverse of "If $ A $, then $ B $" is "If $
eg A $, then $
eg B $". Here, $ A $: two out of five use subway daily, $ B $: probability of a person using subway is 0.4. So $
eg A $: two out of five do not use subway daily, $
eg B $: probability of a person using subway is not 0.4. So option B is "If $
eg A $, then $
eg B $", which is the inverse.

Wait, but let's check the other options. Option C: "If the probability of a person not using the subway is 0.4, then two out of five people don't use the subway every day" – probability of not using is 0.4, so probability of using is 0.6, two out of five not using means three out of five using, proportion 0.6, which matches. But is this the inverse? No, because the original $ B $ is about probability of using being 0.4, not not using. Option D: "If the probability of a person not using the subway is 0.4, then three out of five people use the subway every day" – this is a true statement (since $ 1 - 0.4 = 0.6=3/5 $), but not the inverse of $ p $. Option A is the converse. So the correct option for the inverse is B? Wait, no, maybe I made a mistake. Wait, let's re - express $ p $:

$ p $: Let $ H $: The number of people using the subway out of five is 2. $ C $: The probability that a person uses the subway is $ 2/5 = 0.4 $.

Inverse of $ p $ (If $ H $, then $ C $) is (If $
eg C $, then $
eg H $).

$
eg C $: The probability that a person uses the subway is not 0.4.

$
eg H $: The number of people using the subway out of five is not 2 (i.e., the number of people not using the subway out of five is 2, since total is 5).

So "If the probability that a person uses the subway is not 0.4, then the number of people not using the subway out of five is 2 (i.e., two out of five don't use)". Which is option B: "If two out of five people don't use the subway every day, then the probability of a person using the subway isn't 0.4". Wait, the order is reversed in the option, but logically, "If $
eg C $, then $
eg H $" is equivalent to "If $
eg H $, then $
eg C $"? No, no. The inverse is "If $
eg H $, then $
eg C $"? Wait, no, the inverse of "If $ P $, then $ Q $" is "If $
eg P $, then $
eg Q $". So $ P $: $ H $ (two out of five use), $ Q $: $ C $ (probability 0.4). So inverse is "If $
eg P $ (two out of five do not use), then $
eg Q $ (probability not 0.4)". Which is exactly option B.

# Answer:
B. If two out of five people don't use the subway every day, then the probability of a person using the subway isn't 0.4

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