Exercises due Sep 22, 2020 16:59 PDT

Exercise: Binomial probabilities

0.0/2.0 points (graded)

Recall that the probability of obtainingk

Heads inn

independent coin tosses is(n

k

)p

k

(1p)

nk

, wherep

is the probability of Heads for any given coin toss.

Find the value of

k=0

n

(n

k

)p

k

(1p)

nk

. (Your answer should be a number.)

unanswered

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Exercises due Sep 22, 2020 16:59 PDT

Exercise: Coin tossing

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Use the second method in the preceding segment to find the probability that the 6th toss out of a total of 10 tosses is Heads, given that there are exactly 2 Heads out of the 10 tosses. As in the preceding segment, continue to assume that all coin tosses are independent and that each coin toss has the same fixed probability of Heads. (In this and subsequent questions, your answer should be a number. Do not enter '!' or combinations in your answer.)

unanswered

Exercises due Sep 22, 2020 16:59 PDT

Exercise: Counting partitions

0.0/3.0 points (graded)

We have 9 distinct items and three persons. Alice is to get 2 items, Bob is to get 3 items, and Charlie is to get 4 items.

1. As just discussed, this can be done ina!
2. b!3!4!
4. ways. Finda
5. andb
6. .
7. a=
8. unanswered
11. b=
12. unanswered
15. A different way of generating the desired partition is as follows. We first choose 2 items to give to Alice. This can be done in(c
16. d
17. )
18. different ways. Findc
19. andd
20. . (There are 2 possible values ofd
21. that are correct. Enter the smaller value.)
22. c=
23. unanswered
26. d=
27. unanswered
30. Having given 2 items to the Alice, we now give 3 items to Bob. This can be done in(e
31. f
32. )
33. ways. Finde
34. andf
35. . (There are 2 possible values off
36. that are correct. Enter the smaller value.)
37. e=
38. unanswered
41. f=
42. unanswered

Verify that the answer from part 1 agrees with the answer that you get by combining parts 2 and 3.

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Exercises due Sep 22, 2020 16:59 PDT

Exercise: Counting

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You are given the set of letters{

A, B, C, D, E}

.

1. How many three-letter strings (i.e., sequences of 3 letters) can be made out of these letters if each letter can be used only once? (In this and subsequent questions, your answer should be a number. Do not enter '!' or combinations in your answer.)
2. unanswered
6. How many subsets does the set{
7. A, B, C, D, E}
8. have?
9. unanswered
13. How many five-letter strings can be made if we require that each letter appears exactly once and the letters A and B are next to each other, as either "AB" or "BA"? (Hint:Think of a sequential way of producing such a string.)