Consider a tournament of n contestants in which the outcome is an ordering of these contestants, with ties allowed. That Is. the outcome partitions the players into groups. with the first group consisting of the players who tied for first place, the next group being those who tied for the next-best position, and so on. Let N(n) denote the number of different possible outcomes. For instance, N ( 2 ) = 3 , since, in a tournament with 2 contestants, player 1 could be uniquely first, player 2 could be uniquely first, or they could tie for first a. List all the possible outcomes when n = 3 . b. With N(0) defined to equal 1, argue, without any computations, that N ( n ) = ∑ i = 1 n N ( n − i ) Hint: How many outcomes are there in which i players tie for last place? c. Show that the formula of part (b) is equivalent to the following: N ( n ) = ∑ i = 1 n ( n i ) N ( i ) d. Use the recursion to find N(3) and N(4).
Consider a tournament of n contestants in which the outcome is an ordering of these contestants, with ties allowed. That Is. the outcome partitions the players into groups. with the first group consisting of the players who tied for first place, the next group being those who tied for the next-best position, and so on. Let N(n) denote the number of different possible outcomes. For instance, N ( 2 ) = 3 , since, in a tournament with 2 contestants, player 1 could be uniquely first, player 2 could be uniquely first, or they could tie for first a. List all the possible outcomes when n = 3 . b. With N(0) defined to equal 1, argue, without any computations, that N ( n ) = ∑ i = 1 n N ( n − i ) Hint: How many outcomes are there in which i players tie for last place? c. Show that the formula of part (b) is equivalent to the following: N ( n ) = ∑ i = 1 n ( n i ) N ( i ) d. Use the recursion to find N(3) and N(4).
Consider a tournament of n contestants in which the outcome is an ordering of these contestants, with ties allowed. That Is. the outcome partitions the players into groups. with the first group consisting of the players who tied for first place, the next group being those who tied for the next-best position, and so on. Let N(n) denote the number of different possible outcomes. For instance,
N
(
2
)
=
3
, since, in a tournament with 2 contestants, player 1 could be uniquely first, player 2 could be uniquely first, or they could tie for first
a. List all the possible outcomes when
n
=
3
.
b. With N(0) defined to equal 1, argue, without any computations, that
N
(
n
)
=
∑
i
=
1
n
N
(
n
−
i
)
Hint: How many outcomes are there in which i players tie for last place?
c. Show that the formula of part (b) is equivalent to the following:
N
(
n
)
=
∑
i
=
1
n
(
n
i
)
N
(
i
)
Suppose that there are n = 2k teams in an eliminationtournament, where there are n∕2 games in the first round,with the n∕2 = 2k−1 winners playing in the second round,and so on. How many rounds are in the elimination tournament when there are 32 teams?
A certain type of digital camera comes in either a 3-megapixel version or a 4-megapixel version. A camera store has received a shipment of 15 of these cameras, of which 6 have a 3-megapixel resolution. Suppose that 5 of these cameras are randomly selected to be stored behind the counter; the other 10 are placed in a storeroom. Let X= the number of 3-megapixel cameras among the 5 selected for behind-the-counter storage.
a. What kind of a distribution does X have (name and values of all parameters)?
b. Compute P(X=2), P(X≤2), and P(X≥2).
c. Calculate the mean value and standard deviation of X.
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