Question

6.Geometry John is playing around with probabilities and thinking about rolling a die until he gets a 6. He computes that (5/6)4 is equal to 0.4823, and concludes, “Cool, I’ve got a better than 50% chance of seeing my first six by my fourth roll.” Then Geometry John wonders, “if I roll a die until I roll two sixes, do I have a better than 50% chance of seeing my second six by my eighth roll?” What is the answer to Geometry John’s question? Compute the probability.

7. Suppose you have a shipment of 100 drills, and four of them are defective. If you randomly select ten of the 100 drills and inspect them closely, what is the probability that you will find (at least) one of the defective ones in your sample of ten?

9. If you call a randomly selected person, there is a 1/10 chance that they are left-handed. If you sequentially call randomly selected people until you speak to your third left-handed person, what is the probability that it this takes exactly 10 calls?

please help me to solve this questions

Answer 1

6.

P(6) = 1/6

P(not 6) = 5/6

P(no. of 6 = x in n rolls) = (nCx)*((1/6)^x)*((5/6)^(n-x))

P(2 sixes by 8 rolls) = 1 - P(0 sixes in 8 rolls) - P(1 six in 8 rolls)

= 1 - (8C0)*((1/6)^0)*((5/6)^(8-0)) - (8C1)*((1/6)^1)*((5/6)^(8-1))

= 1 - 1*((1/6)^0)*((5/6)^(8-0) - 8*((1/6)^1)*((5/6)^(8-1))

P(2 sixes by 8 rolls) = 0.3953

7.

P(no defective in 10 selected) = (no of ways to select from 96 non defective)/(no. of ways to select from 100 drills)

= (96C10)/(100C10)

= (11,279,926,456,656) / (17,310,309,456,440)

P(no defective in 10 selected) = 0.6516

P(atleast 1 defective) = 1 - P(no defective in 10 selected)

= 1 - 0.6516

P(atleast 1 defective) = 0.3484

8.

P(person is left handed) = 0.1

P(person is not left handed) = 1-0.1 = 0.9

P(no. of left handed person = x in n calls) = (nCx)*((0.1)^x)*((0.9)^(n-x))

if it takes exactly 10 calls it means the 10 th person was left handed and 2 left handed person in previous 9 calls

P(no. of left handed person = 2 in 9 calls) = (9C2)*((0.1)^2)*((0.9)^(9-2)) = 36*0.0047 = 0.1692

P(3rd left handed person in 10 th call) = P(no. of left handed person = 2 in 9 calls)*P(10th person is left handed)

= (0.1692)*(0.1) = 0.01692

P(3rd left handed person in 10 th call) = 0.01692

P.S. (please upvote if you find the answer satisfactory)