This is the probability that the passenger obtains a boarding pass which

is 1 minus 0.5.

Times the probability that the passenger is not caught by border patrol,

which is 1 minus 0.9.

Times the probability that the passenger is not caught at the gate,

which is 1 minus 0.3.

This simplifies to 0.5 times 0.1 times 0.7,

which equals 0.035, or 3.5%.

The second question asks,

what the probability a guilty passenger actually boards the plane is?

This is the probability that the passenger makes it past the gate,

times the probability that the passenger makes it past the fourth checkpoint.

That is, 0.035 times 0.95 for

the 95% chance the passenger is not selected for

screening, plus 0.05 times 1 minus 0.99.

For the 5% chance the passenger is selected for

screening, times the probability the violation is caught.

Notice that I have two sets of parenthesis here so

we need to be mindful of order of operations.

This simplifies to 0.035 times

0.95 plus 0.05 times 0.01,

which is 0.0332675, or about 3.3%.