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Probability & Statistics/Probability of two things happening at the same time

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QUESTION: Hi! I have a probability question about two things happening at the same time.

Let's say that I randomly turn on my TV 10 times every day but just for 10 minutes each time. (Most of the time I am NOT watching TV.) My home power grid fails once every day, at a random time. What is the probability of the power failing WHILE I am watching TV? Thanks!

ANSWER: Your TV is on for 100 minutes out of 1440 in a day.

If the power outage is truly random, the probability it happens in those 100 minutes is 100/1440, which is slightly less than 7%.

---------- FOLLOW-UP ----------

QUESTION: Thanks for the reply! I have a follow up question. I thought that 100/1440 = 6.9% would be the probability that I am watching TV at any point in the day.
Wouldn't the probability of a power outage during those 100 minutes be 1/1440 = 0.069%? What if the power goes out twice a day? Thanks!

Answer
You're right -- for every minute you watch TV, the probability of the outage during that minute is 1/1440. But you watch for 100 minutes, so they accumulate and the total probability each day becomes 100/1440.

If the two outages are each random (not dependent at all) then this is a slightly different problem.

The probability of NOT seeing outage #1 is 1340/1440.

The probability of NOT seeing outage #2 is 1340/1440.

The probability of seeing NEITHER is (1340/1440) (1340/1440).

The probability of NOT seeing NEITHER (i.e. seeing one of them) is:

1 - (1340/1440) (1340/1440) ≈ 13.4%

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Clyde Oliver

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I can answer all questions up to, and including, graduate level mathematics. I do not have expertise in statistics (I can answer questions about the mathematical foundations of statistics). I am very much proficient in probability. I am not inclined to answer questions that appear to be homework, nor questions that are not meaningful or advanced in any way.

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I am a PhD educated mathematician working in research at a major university.

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AMS

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Various research journals of mathematics. Various talks & presentations (some short, some long), about either interesting classical material or about research work.

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BA mathematics & physics, PhD mathematics from a top 20 US school.

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Various honors related to grades, various fellowships & scholarships, awards for contributions to mathematics and education at my schools, etc.

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In the past, and as my career progresses, I have worked and continue to work as an educator and mentor to students of varying age levels, skill levels, and educational levels.

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