Question

burgular alarm system has six fail-safe components. the probability of each failing is 0.05. Find these probabilities a)Exactly three will fail b) Fewer than two will fail. c) None will fail. d) Compare the answers from a,b,and c and explain why the results are reasonable

Answer 1

a)\[\left(\begin{matrix}6 \\ 3\end{matrix}\right)*.5^6\]

Answer 2

for part b you would just want to add the probablility none will fail to the probability that only 1 will fail

Answer 3

i still dont get part b

Answer 4

if you want the chance that less than two will fail then there are 2 situations that make that possible. If none of them fail, or if only one fails.

the probablility that only one fails would be \[\left(\begin{matrix}6 \\ 1\end{matrix}\right)(.5^1)(.5^5)\]

the probability that none fail would be (.5)^6

you just need to get those two probabilities and add them together to find the chance that less than two fail

Answer 5

so why are all of them reasonable

Answer 6

haha. well what do you think... why is the probability that 1 fails bigger than the probability that none fail?

Answer 7

so say the probability that none fail would be (.5)^6 ...but that would mean that the chance that not a single one fails get smaller by increasing the number of the components? it should get bigger shouldnt it?

-> probability that one fails= (.5)^1 < (.5)^6 =probability that none fail

what am I not getting here?

Answer 8

sorry for the late response... the reason that the probability of one failing is bigger than the probability of none failing is that there are more ways for only one to fail. look at it this way...

lets say that S represents success and F represents failure...

if we want all to succeed then we must get

S S S S S S

if we want one to fail then we can get

F S S S S S OR
S F S S S S or
S S F S S S
S S S F S S
S S S S F S
S S S S S F

thats why the chances of 1 failing are 6 times bigger than the chances of none failing