163 Discrete Mathematics Review 2 Use the following to answer ...

Use the following to answer question 8:
In the questions below an experiment consists of picking at random a bit string of length five.
Consider the following events:
E 1 : the bit string chosen begins with 1;
E 2 : the bit string chosen ends with 1;
E 3 : the bit string chosen has exactly three 1s.
8. Determine whether E 2 and E 3 are independent.
Solution. Let us first find the probability of
. The total number of bit strings of length
5
4
5 is 2 = 32. The number of strings ending with 1 is 2 = 16. Therefore,
16 1
C(5,3) 5
pE (
2)
= = . In a similar way probability of E3
is PE (
3)
= = . Finally let
5
32 2
2 16
us compute the probability of E ∪ E . The number of strings ending with 1 and having
2
3
6 5
exactly three 1s is C (4,2) = 6 whence PE (
2
∪ E3)
= ≠ pE (
1) pE (
3)
=
32 32
are not independent.
E 3
E 2
and E and 2
9. Each of 26 cards has a different letter of the alphabet on it. You pick one card at
random. A vowel is worth 3 points and a consonant is worth 0 points. Let X = the value
of the card picked. Find E(X), V(X), and the standard deviation of X.
5 21 1
Solution. EX ( ) = 3⋅ + 0⋅ = 5 .
26 26 26
2 2
⎛ 15 ⎞ 5 ⎛ 15 ⎞ 21
V( X) = ⎜3− ⎟ ⋅ + ⎜0 − ⎟ ⋅ ≈1.40
.
⎝ 26 ⎠ 26 ⎝ 26 ⎠ 26
σ ( X) = V( X) ≈ 1.18
Use the following to answer question 10:
In the questions below, describe each sequence recursively. Include initial conditions and
assume that the sequences begin with a 1 .
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