Draw the Probability Tree Model Discrete Structures Questions

1.A box contains three coins: two regular coins and one fake two-headed coin. You pick a coin at random and toss it. What is the probability that it lands heads up?

2. A committee of 5 people is to be chosen from a group of 9 women and 6 men. If the selection is made randomly, what is the probability that the committee will consist of 2 women and 3 men?

3. An urn contains n balls, one of which one is special. If k of these balls are withdrawn one at a time, with each selection being equally likely to be any of the balls that remain at the time, what is the probability that the special ball is chosen?

4. A system is made up of 5 components, each of which is either working (w), or failed (f). Consider an experiment consisting of observing the status of each component and let the outcome of the experiment be given by a vector (x1,x2,x3,x4,x5) where xi is equal to 1 if the component is working and is equal to 0 if component i is failed.

a)How many outcomes are in the sample space of this experiment?

b) Suppose that the system will work if components 1 and 2 are both working, or if components 3 and 4 are both working, or if components 1, 3, and 5 are all working. Let W be the event that the system will work. Specify all the outcomes in W……note that it will be a (long) set containing the vectors representing the working conditions.

c) Let A be the event that components 4 and 5 are both failed. How many outcomes are contained in the event A?

d) Write out all the outcomes in the event A^W.

5. If you have 2 coins, with one being fair and the other having two heads, and you then pick one of the two coins at random with equal probability out of an urn and without looking at both sides to see whether it is fair or not, and then flip it to determine whether you get heads or tails, and then repeating such a process 100 times, for a total of 100 flips.

a) Compute the probabilities of getting exactly 60 heads.

b) Generalize the result for getting exactly m heads after n flips.

6. A toy company produces two kinds of model cars – 40% of which are sedans and 60% of which are SUVs. They come in the two colors: red or blue. The sedans are 50% red and the SUVs are 75% red. After production the toy cars are randomly partitioned into boxes with 100 cars per box. You buy one of the boxes.

a) Draw the probability tree model for the model car manufacturing process and label only the branch probabilities involved (do not calculate the leaf or terminal node)

b) What is the probability that you choose a blue car from the box?

c) What is the probability that the blue car that you chose in part b) is an SUV?