W03 - HW Solutions

01

(1) Define events.

Let be the event that Jim is guilty.

Let be the event in which the DNA matches.

We are given that .

We know that .

Since there are 100,000 citizens, , .

We are asked to compute .

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(2) Use Bayes’ Theorem to set up the formula for .

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(3) Plug in values.

02

(1) Define events.

Let be the event where the Winning Fund outperforms the first year.

Let be the event where the Winning Fund outperforms the second year.

We are asked to compute .

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(2) Find relevant probabilities.

.

.

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(3) Compute

03

Use the multinomial coefficient.

Use that .

We have three bins, , , and .

Thus,

04

(a)

(1) Consider total number of outcomes.

You roll a die times, so there are total outcomes.

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(2) Consider total number of desired outcomes.

We choose 16 of the 100 to be ones.

We choose 17 of the remaining 100 - 16 = 84 to be twos.

The other can be any of . So, there are remaining outcomes.

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(3) Set up formula.

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(b)

(1) Consider total number of desired outcomes.

We choose 25 of the 100 to be ones.

Out of the remaining 100 - 25 = 75 rolls, we choose at least 50 to be two.

The remaining rolls, where , can be any of , , , . So, there are remaining outcomes.

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(2) Set up formula.

05

(1) State the formula for a binomial distribution.

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(2) State parameters of binomial distribution.

since you buy paper clips.

ranges from to .

.

.

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(3) Use summation notation to find .

06

(1) Describe the situation.

Let .

If it takes at least rolls for a to come up, then the first rolls resulting in a number that is not a six.

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(2) Compute relevant probabilities.

The probability that you roll a six is .

The probability that you do not roll a six is .

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(3) Set up formula.

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(4) Use the formula for geometric series to evaluate the sum.

07

(a)

The odds of a perfect week follows a binomial distribution.

since there are days to account for.

We choose of them to have accidents.

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(b)

(1) The odds of exactly 2 accidents in 30 days follows a binomial distribution.

since there are 30 days to account for.

We choose of these days to have accidents.

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(2)

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(c)

(1) Describe the situation.

We want four perfect days before an accident occurs.

Over the course of the next six days, we want at least one accident occurring.

Since these are independent trials, multiplying the individual probabilities together will yield the desired value.

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(2) Set up the first condition.

since we want to account for 4 days.

We choose of these days to have accidents.

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(3) Set up second condition.

since we now want to account for the next six days.

We choose of these days to have accidents, where ranges from through .

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(4) Multiply the two expressions.

08

(1) The probability that you pass follows a binomial distribution.

since there are 4 questions to account for.

We choose questions to be correct, where or .

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(2) Describe the situation wherein you get the second question right.

Now, out of the remaining three questions, you need to answer at least two correctly.

This too follows a binomial distribution.

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(3) Set up binomial distribution formula.

, since we only need to account for the remaining 3 questions.

We choose questions to be correct, where or .

09

(1) Describe dependencies of components.

The series 4, 5 runs parallel to component 3.

The components 2, 3, 4, 5, 6 all run in a series.

The latter is parallel to component 7

The circuit starts at component 1 and ends at component 8

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(2) Find the probability the component 4 and 5 succeeded. (Denote as )

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(3) Find the probability 3, and succeed. Note that these are in parallel. (Denote as )

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(4) Find the probability 2, , and 6 succeed. (Denote as )

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(5) Find the probability 7 and succeed. Note that these are in parallel. (Denote as )

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(6) Lastly, find the probability , , and 8 succeed. (Denote as ).

10

(1) Find the possible values of .

After the decimal point, any of the 10 digits can appear, so .

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(2) Find the PMF of .

Each digit has a chance of appearing. So, the PMF of is

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(3) Find the CDF of .

The CDF of is given as follows.

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(4)

Listing out the individual cumulative probabilities for each is also acceptable.

11

(a)

Describe the situation.

Let be A’s (not total) winnings after the round.

Since A either loses one dollar or gains one dollar each round, .

, so .

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(b)

(1) Describe the PMF of .

if and only if all 5 rounds are tails, so .

if and only if all 5 rounds are heads, so .

if 4 rounds are tails, and if 4 rounds are heads. Since the coin is fair, .

if 3 rounds are heads, and if 3 rounds are tails. Since the coin is fair, .

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(2) Define the PMF of .

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(3) Define the CDF of .

Note that the jumps in the PDF are the possible discrete values of .

12

13

(a)

(b)

14

15

16

(a)

(b)

17

(a)

(b)

18

First find the PMF of :

	0	1	2	3
	0.1	0.1	0.3	0.5

Now the PMF of :

	0	3	6	9
	0.1	0.1	0.3	0.5