Need help with joint distribution problems

#1
1. You need $30,000. Suppose that there is a 20% chance that Bank of
America will loan you $30,000, a 30% chance that Wells Fargo will loan you $20,000, and a 40% chance that Washington Mutual will loan you $10,000. What is the probability that you will be able to borrow enough money?


What I did: There's a 1 - (1-0.2)(1-0.3)(1-0.4) = 66% chance that at least one of those banks will lend you money. If $10k is enough, then that'll do. But I don't know how to find out the probability of being able to borrow enough money (30k)...

2. A bundle of 40 t-shirts are delivered to the store. They randomly select 4 t-shirts for inspection. If none or one is defective, the bundle is accepted. If two or more are
defective, the bundle is rejected. What is the probability that the bundle will be rejected if only 10% of the bundle is defective?


What I did: So 4 t-shirts are defective, and 36 are not. The probability of getting a defective shirt is 4/40, and the probability of not getting an effective shirt is 36/40. The probability that the shirt is accepted is 37/40, and the probability that a shirt is rejected is 2/40?
I am having such a hard time with these problems, can you guys show me a step-by-step process on how I can arrive to the answer?


Thanks!
 
#3
1. Step by step

... BOA WF WM
Yes 0.2 0.3 0.4
.No 0.8 0.7 0.6

These are the 8 possible outcomes, the last column shows total money raised

BOA WF WM
Yes Yes Yes $60,000.00
Yes Yes No $50,000.00
Yes No Yes $40,000.00
Yes No No $30,000.00
No Yes Yes $30,000.00
No Yes No $20,000.00
No No Yes $10,000.00
No No No $0.00

So the first 5 cases raise $30,000 or more:

(BOA Yes) and (WF Yes) and (WM Yes) = 0.2 * 0.3 * 0.4 = 0.024
(BOA Yes) and (WF Yes) and (WM No ) = 0.2 * 0.3 * 0.6 = 0.036
(BOA Yes) and (WF No ) and (WM Yes) = 0.2 * 0.7 * 0.4 = 0.056
(BOA Yes) and (WF No ) and (WM No ) = 0.2 * 0.7 * 0.6 = 0.084
(BOA No ) and (WF Yes) and (WM Yes) = 0.8 * 0.3 * 0.4 = 0.096

Adding up those probabilities, you get 0.296

Or what BGM said

(BOA Yes) or (BOA No and WF Yes and WM Yes)
=(0.2) or (0.8 and 0.3 and 0.4)
=(0.2) + (0.8 * 0.3 * 0.4)
=0.296
 
Last edited: