EVENT PROBABILITY

First set of O- rings function
correctly and second O-ring
functions correctly .7552

First set of O-rings malfunction
and second O-ring functions
correctly .1138

1st set of O-rings function
correctly and 2nd O-ring
malfunctions .1138

1st set of O-rings malfunction
and 2nd O-ring malfunctions .0172

Fill in the probability distribution table

x P(x)
0
1
2
Total

Then using the probabilities from the table find

P(Explosion on lift-off)

P(Safe lift-off)

To fill in the probability distribution table, we need to calculate the probabilities for each possible outcome.

Let's define the events as follows:
- First set of O-rings functioning correctly as A
- Second O-ring functioning correctly as B

Now let's calculate the probabilities for each outcome:

x = 0 (Both O-rings malfunction):
P(x = 0) = P(A' ∩ B') = P(A) × P(B) = (1 - 0.7552) × (1 - 0.7552) = 0.0172

x = 1 (One O-ring malfunctions):
P(x = 1) = P((A' ∩ B) ∪ (A ∩ B')) = P(A') × P(B) + P(A) × P(B') = (1 - 0.7552) × 0.7552 + 0.7552 × (1 - 0.7552) = 0.1138 + 0.1138 = 0.2276

x = 2 (Both O-rings function correctly):
P(x = 2) = P(A ∩ B) = P(A) × P(B) = 0.7552 × 0.7552 = 0.5707

To find the total probability, we sum up the probabilities for all possible outcomes:

Total = P(x = 0) + P(x = 1) + P(x = 2) = 0.0172 + 0.2276 + 0.5707 = 0.8155

Now let's calculate the remaining probabilities:

P(Explosion on lift-off):
The explosion can occur when both O-rings malfunction, so the probability is equal to P(x = 0) = 0.0172.

P(Safe lift-off):
A safe lift-off can occur when at least one O-ring functions correctly, so the probability is equal to P(x = 1) + P(x = 2) = 0.2276 + 0.5707 = 0.7983.

So, the probability of an explosion on lift-off is 0.0172 (or 1.72%), and the probability of a safe lift-off is 0.7983 (or 79.83%).