A super has a key ring with 12 keys. He has forgotten which one opens the apartment he needs to enter. What is the probability h
1 answer:
Answer:
Therefore, the probability is P=1/4.
Step-by-step explanation:
We know that a super has a key ring with 12 keys. He has forgotten which one opens the apartment he needs to enter.
We conclude that each key has an equal probability of opening an apartment. Since there are 12 keys, it follows that the probability for each key is equal,
p = 1/12.
We calculate the probability he is able to enter before reaching the 4th key. So he will try three keys by then. We get:
Therefore, the probability is P=1/4.
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For the remaining limit, use the Taylor expansion for cos(x) :
where essentially means that all the other terms in the expansion grow as quickly as or faster than x⁴; in other words, the expansion behaves asymptotically like x⁴. As x approaches 0, all these terms go to 0 as well.
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Unfortunately, this doesn't agree with the limit we want, so n ≠ 3. But you can try applying this method for larger n, or computing a more general result.
Edit: some scratch work suggests the limit is 10 for n = 6.