Math Is Fun Forum

A square mesh of wires has a diameter d, and the holes in the mesh are squares whose side length is w. A spherical particle of radius r is dropped on the mesh. What is the probability that it passes through? What is the probability that it fails to pass through if it is dropped n times?
a)
f(x,y) + {1/r, if x^2 + y^2 <= 1 and 0 otherwise.
It would be r^2
How would I solve this? would I find the marginal density?

b)
I was thinking of using binomial distribution to find the probability, since its talking about failure. So, would it be like this?
P(N, n) = (n C N ) * p ^ k * (1-p)^(N-n) would I just show this?

Thanks for your time.

Catherine would like to invite at least 5 of her friends to a party. Suppose each of her friends, if invited, independently agrees to come with probability 3/4. What is the probability that she will get 5 to agree to come if she invites 5 people?
The answer is (5 C 5) (0.75) ^5(0.25) ^0. So, where did the solution get 0.25???? Is there a formula for this?? Thanks

A system runs three shifts. In a day, 1% of the items produced by the first shift are defective, 2% of the second shift's items are defective, and 5% of the third shift's items are defective. If the shifts all have the same productivity, what percentage of the items produced in a day are defective?
1/100 * 2/100 * 5/100 = 10/100 = 1/10, so 10% percent defective right?

If an item is defective, what is the probability that it was produced by the 3rd shift?
2% + 1% + 5% = 8% total, 5% / 8% = 0.625

Are these answers correct? Thank you for your time