Carcass wrote:
A string of 10 lightbulbs is wired in such a way that if any individual lightbulb fails, the entire string fails. If for each individual lightbulb the probability of failing during time period T is 0.06, what is the probability that the string of lightbulbs will fail during time period T ?
A. \(0.06\)
B. \((0.06)^{10}\)
C. \(1 - (0.06)^{10}\)
D. \((0.94)^{10}\)
E. \(1 - (0.94)^{10}\)
Aside: If P(bulb fails) = 0.06, then P(bulb doesn't fail) = 0.94Okay, the entire string of lightbulbs will fail if 1 or more lightbulbs fail.
So, we want P(
at least 1 lightbulb fails)
When it comes to probability questions involving "at least," it's best to try using the complement.
That is, P(Event A happening) = 1 - P(Event A
not happening)
P(at least 1 lightbulb fails) = 1 -
P(zero lightbulbs fail)P(zero lightbulbs fail)P(zero lightbulbs fail) = P(1st bulb doesn't fail
AND 2nd bulb doesn't fail
AND 3rd bulb doesn't fail
AND . . .
AND 9th bulb doesn't fail
AND 10th bulb doesn't fail)
= P(1st bulb doesn't fail)
x P(2nd bulb doesn't fail)
x P(3rd bulb doesn't fail)
x . . .
x P(9th bulb doesn't fail)
x P(10th bulb doesn't fail)
= (0.94)
x (0.94)
x (0.94)
x . . .
x(0.94)
x (0.94)
=
(0.96)^10So, P(at least 1 lightbulb fails) = 1 -
P(zero lightbulbs fail)= 1 -
(0.94)^10Answer: E
Cheers,
Brent