Binomial distributions, standard deviation and more
Hi guys, I need your help with a few probability problems. I'm very new to the topic so even though the questions may seem very easy to you, they're very new to me. So, if you can, I'm thankful for every help you can give me.
My first problem: A diamond worth $20000 is covered by an insurance against loss and theft. In case of a theft, the insurance pays the full price. In case of a loss, the insurance pays $8000. Assuming that the probability of theft is 0.0025 and of loss 0.03, how much should the insurance company charge to cover the ring, if they want $100 expected return?
Problem 2: In a company, 35% of the workers work at night. Seven workers are randomly selected, what is the probability that i:3 ii: less than 4 iii: at least 4 of them work at night?
Problem 3: In a cake factory, 6 out of 100 cakes are too sweet. 12 cakes are selected randomly (with replacement). What is the probability that
I: none ii: maximally 1 iii: at least 2 iiii: less than 4 will be too sweet?
Problem 4: A student is very often late. The probability that he is late is 2/5. On the weekend, he has also some special classes. What is the probability that in a random week he is on time: i: 7 days ii: Tuesday iii: any 6 days iiii: at least 4 days
Last problem: A company fills bottles with water. The probability that the machines do not fill a bottle with as much water as they should is 4%. 30 Bottles are randomly chosen. X is the number of bottles that are not filled properly. What is the mean and the standard deviation?
Thank you very much for your help on these problems. For the ones with more parts to it, it will already help me a lot if you help me with the first 2 or so and explain to me how to do them, then I can hopefully calculate the other parts on my own, but, of course, if you can help me with all, that would be even greater. Already in advance, thank you for your effort.