Question

Problem 3 (Systems Engineering) The Challenger was a space shuttle that exploded on January 28, 1986 shortly after take off. The cause of the accident was found to be a gasket, called an O-ring, that was designed to act as a seal on the booster rockets. For simplicity, we'll assume there were 2 9rinss per booster rocket (there were actually 3) and two booster rockets on the space shuttle. Both booster rockets must function for the space shuttle to launch successfully, but only one of the o-rings was required to be functional for the booster rocket it was inside to work g-rings was required to be functional Historical data collected from previous launches showed that an individual o ringfaled on about 30%of the launches. Assuming that the g-rings were identical and operated independently i.e failure of one 9ring did not affect whether another g-ring would fail), what was the probability that a space shuttle would have a catastrophic failure? (Hint: draw a "systems" diagram like we did before with the booster rockets and the o- rings inside each booster rocket to be in either a parallel or series configuration a b) Historical data collected showed that on launches at or below 70*F, the probability a single O-ring would fail was about 66%, when the space shuttle launched, the temperature was considerably lower than 70 degrees. Assuming that under the conditions the Challenger was launched, the probability a single gring failed was 66%, what would NASA have found was the probability the Challenger would explode? c) Morton Thiokol, who made the O-rings, came under great scrutiny after the disaster. Suppose you were an engineer for NASA back in 1986 and you drew up new guidlelines for reducing disasters due to o-ring failure. What should you require the probability a single O-ring will fail if you want the probability of a catastrophic space shuttle launch to be less than 1%?

Answer 1

Solution

Back-up Theory

Let p be the probability a O-ring failing. Then, given

‘two O-rings per booster rocket and that only one required to be functional for the booster rocket it was inside to work’

=> probability of a booster rocket working

= P(at least one O-ring functions)

= 1 - P(neither O-ring functions)

= 1 – p², [given O-rings are independent] …..…………………………………………………. (1)

Assuming 2 booster rockets in the space shuttle and given both must function for the successful launch of the space shuttle,

P(successful launch) = [1 – p²]² ……………………………………………………………… (2)

Now to work out the solution,

Part (a)

Here p = 0.3 [given that an individual ring failed 30% of the time]. Then vide (2),

P(successful launch) = [1 – 0.3²]²

= 0.8372

=> P(catastrophic failure)

= 1 – 0.8372

= 0.1628 Answer

Part (b)

Here p = 0.66

P(successful launch) = [1 – 0.66²]²

= 0.3185

=> P(failure or explosion)

= 1 – 0.3185

= 0.6815 Answer

Part (c)

We want P(failure) < 0.05

=> P(successful launch) > 1 – 0.05 = 0.95

i.e., [1 – p²]² > 0.95 [vide (2)]

or 1 – p² > 0.9950

=> p² < 0.0050

Or, p < 0.0707

Thus, the probability of a single O-ring failing should be less than 7% Answer

DONE