Jobs Simply

Help with Probability Trees?

I don't know how to make the trees for the following questions and solve them, help would be great -.- got an exam comming up on probabilty soon so all the help would be appreciated! 1) A bag contains three red and two black marbles. One marble is selected at random, the colour noted an treturned. A exond selection is made in the same way. Fine the probability of selecting: a) Two red marbles b) a red marble followed by a black marble (in that order) c) Two different coloured marbles 2) A bag contains three red and two black marbles. One marble is selected at random, the colour oted and not replaced. A second selection is then made. Find the probability of selecting: a) Two red marbles b) A red marble and a black marble in that order c) Two different coloured marbles. 3) A duo is selected from four baritones and two sopranos. Find the probability that the duo is made up of: a) two baritones b) two sopranos c) a baritone and a soprano 4) On my bookshelf there are fifteen novels and eight maths books. I select one book at random, not its type and return it I select another book in the same way. Find the probabilty of choosing: a) a novel then a maths book b) two novels c) at least one maths book. 5) A bag contains 8 red disc and 11 black discs. A disc is selected at random, its colour noted and returned. A second disc is selected. Find the probability of selecting: a) two red discs b) two black discs c) a black disc first then a red disc d) different coloured discs 6) On my bookshelf are twelve novels and nine science books. I select a book at random, not its type and do not return it. I select another book in the same way. Find the probability of choosing: a) a novel then a science book b) two novels c) atleast one science book 7) A bag contains eight red discs and eleven black discs. A disc is selected at random, its colour noted but not returned. A sexon disc is selected. Find the probability of choosing: a) two red discs b) two black discs c) a black disc first then a red disc d) different coloured discs. 8) There are ten men and seven women on a committee. Five of the men and three of the women have previous experince at their job. If two people are selected at random from this group, find the probability that: a) they are both female b) they have both had previous experince c) they are both male without previous experince.

Public Comments

1. a probability tree is just a way of expressing all the diff scenarios with their probability of occurring. you always start from a point, and it gets more complex along the way. i shall use a simple example before i answer your specific questions and hope it helps you understand probability trees. as an example, lets consider the gender and race of a newborn baby. let the probability of a baby being male or female be 1/2 each. assume the country in question has 3 different races (A B C), with the population size and birth rate of each race making the probability that a random baby is of race A is 1/6, B is 1/3, C is 1/2. (notice that they must add up to 1) you start with a dot as your starting point. now with 2 factors(gender/race), you have 2 levels of probability tree, and you can either go race-gender or gender-race (this usually depends on what probability the question give you). the important thing is that for each dot (either the starting or the intersections), all the branches that fan out directly from it (not indirectly) must add up to 1 in their probability. for race-gender, you branch out into 3 branches to represent each race, then scribble the probability for each race above the branch. then for each of the 3 end-points you have now, you draw 2 branches, with 1/2 written on each branch. for gender-race, you branch out 2 branches from the starting dot, and write 1/2 on each. then you branch out 3 branches from each end-point and write out the individual probabilities. either way, you will end up with 6 end-points. oh, and to use your probability tree, simply multiply level 1 and level 2 as you trace from starting dot to end-point. the probability of a baby boy of race A being born is 1/12, and a baby girl of race C is 1/4. ___ OKAY now lets move on to answering your question. i sure hope the above helps though ^^ probability trees arent that hard. 1) you should use a probability tree with 1-2-4. probability of red is 3/5 and black is 2/5 because they are equally likely to be chose since its random picking, hence the probability is just their percentage in the mixture. since the first marble is returned, the second selection has the same probabilities. a) Two red marbles: (3/5)*(3/5) = 9/25 b) a red marble followed by a black marble, in that order: (3/5)*(2/5) = 6/25 c) Two different coloured marbles: (3/5)*(2/5) + (2/5)*(3/5) = 12/25 2) this is similarly 1-2-4 structure for the probability tree but note that the first marble isnt replaced hence the probabilities are now different. first marble has 3/5 chance of being red and 2/5 chance of being black. now consider the ratio in the remaining 4 marbles for second marble. if the first marble is red, there's 2 red and 2 black left, so 2/4 for either colours. if the first is black, there's 3 red and 1 black left so 3/4 and 1/4 depending on the colour. a) Two red marbles: (3/5)*(2/4) = 3/10 b) A red marble and a black marble, in that order: (3/5)*(2/4) = 3/10 c) Two different coloured marbles: (3/5)*(2/4) + (2/5)*(3/4) = 3/5 3)this question is also 1-2-4. a) two baritones: (4/6)*(3/5) = 2/5 b) two sopranos: (2/6)*(1/5) = 1/15 c) a baritone and a soprano: (4/6)*(2/5) + (2/6)*(4/5) = 8/15 4) 1-2-4. there's replacement so it should be easier. a) a novel then a maths book: (15/23)*(8/23) = 120/529 b) two novels: (15/23)*(15/23) = 225/529 c) at least one maths book: (15/23)*(8/23) + (8/23)(15/23 + 8/23) = 304/529 5) 1-2-4. there's replacement so it should be easier. a) two red discs: (8/19)*(8/19) = 64/361 b) two black discs: (11/19)*(11/19)= 121/361 c) a black disc first then a red disc: (11/19)*(8/19)= 88/361 d) different coloured discs: (11/19)*(8/19) + (11/19)*(8/19)= 176/361 6) 1-2-4. no replacement hence diff probabilities. a) a novel then a science book: (12/21)*(9/20) = 9/35 b) two novels: (12/21)*(11/20) = 11/35 c) at least one science book: (12/21)*(9/20) + (9/21)*(12/20 + 8/20) = 24/35 7) 1-2-4. no replacement hence diff probabilities. a) two red discs: (8/19)*(7/18) = 28/171 b) two black discs:(11/19)*(10/18) = 55/171 c) a black disc first then a red disc: (11/19)*(8/18) = 44/171 d) different coloured discs: (8/19)*(11/18) + (11/19)*(8/18) = 88/171 8) this is tricky. the easiest way is to not use the probability tree but reason out. if you really want to practice using the trees, i think the most elegant one would be 1-4-16. you would need to spend some time to work out the individual probabilities. another way would be you can draw 3 different probability trees for the 3 parts, 1-2-4, 1-2-4, 1-4-16. (cant avoid the 1-4-16 one haha) a) they are both female: (7/17)*(6/16) = 21/136 b) they have both had previous experience: (8/17)*(7/16) = 7/34 c) they are both male without previous experience. (5/17)*(4/16) = 5/68