With one alarm clock, we have 0.975 probability of being awakened?

Question

The problem reads:

With one alarm clock, we have 0.975 probability of being awakened. What is the probability of being awakened if we are using two such alarm clocks?


I believe the solution is P (two alarm clocks)=(0.975)^2, or 0.951. I just got worried because 0.951 is lower than 0.975, and shouldn't ur chance of waking up increase if you use two alarm clocks more than one?

Confused

Answer 1

P(not being awaken) = prob of both alarm clocks NOT going off.
You should not square 0.975, but rather (1-0.975) and subtract it from 1.

Ans. 1 - 0.025^2 = 0.999375

Answer 2

Multiplying the probability numbers together can seem like the logical thing to do, but gives us an incorrect result. Let us use the below formula to devise the answer.

P(A ÈB) = P(A) + P(B) - P(A) P(B)

P= probability. A= first alarm clock. B= second alarm clock

P(A ÈB) = P(.975) + P(.975) - P(.975) * P(.975)

P(A ÈB) = P(1.95) - P(.950625)

P(A ÈB) = .999375

What we did above was to add alarm clock one probability to alarm clock two probabilty. Then we multiplied alarm clock one probability to alarm clock two probabilty. After that, we minused the multiplied sum to our added sum, which gives us the .999375 result. In Statistics, common sense is your best defense against incorrect results, as you mentioned above. Hope this helps.

Answer 3

P (2 alarm clocks) = 1 - .975 * 1 - .975 = .025 * .025 = 1 -.000625 = 0.999375

0.99375

Ya it's confusing but ur just taking the odds of both not waking you up and subtracting from 1.

Answer 4

Hello helloworld. This can be confusing. You need to invert the 'risk' to the risk of NOT waking up. This is 1 - 0.975 = 0.025

Now, multiple the risk of not waking up using two alarm clocks: (0.025)(0.025) = 0.000625.

Now convert back to the probability of waking up with two alarm clocks, which is:
1 - 0.000625 = 0.999375

Best wishes and good luck.

Answer 5

You have a higher chance of being awakened with two alarm clocks. Most of the time you should trust your intuition on problems like this.
The first alam goes off you have a .025 chace of remaining asleep. Then the second alarm rings and if your still asleep you have a .025 chance to remain asleep. So .025*.025 chance of sleeping through both alarms.
Or 1-.025^2 chance of waking up

Answer 6

Well, there's a little more to it than that. There are TWO ways you could be woken: (1) The first clock wakes you OR (2) you are not woken by the first and then the 2nd one wakes you. So here we go:
P(woken by 1st alarm clock) = .975
P(NOT woken by 1st, woken by 2nd) = (1-.975)(.975) = .024375

Add the 2 outcomes = .975 + .024375 = .999375

Answer 7

set up as equal porportions 1 is to .975 as 2 is to x

solve for x by cross multiplying.

x=1.95