Answer:
a) 1/2
b) 1/n
c) 1/4
Step-by-step explanation:
a) For each permutation, either 1 precedes 2 or 2 precedes 1. For each permutation in which 1 precedes 2, we can swap 1 and 2 to obtain a permutation in which 2 preceds 1. Thus, half of the total permutations will involve in 1 preceding 2, hence, the probability for a permutation having 1 before 2 is 1/2.
c) If 2 is at the start of the permutation, then it is impossible for 1 to be before 2. If that is not the case, then 1 has a probability of 1/n-1 to be exactly in the position before 2. We can divide in 2 cases using the theorem of total probability,
P( 1 immediately preceds 2) = P (1 immediately precedes 2 | 2 is at position 1) * P(2 is at position 1) + P(1 immediately precedes 2 | 2 is not at position 1) * P(2 is not at position 1) = 0 * 1/n + (1/n-1)*(n-1/n) = 1/n.
d) We can divide the total of permutations in 4 different groups with equal cardinality:
- Those in which n precedes 1 and n-1 precedes 2
- those in which n precedes 1 and 2 precedes n-1
- those in which 1 precedes n and n-1 precedes 2
- those in which 1 precedes n and 2 precedes n-1
All this groups have equal cardinality because we can obtain any element from one group from another by making a permutations between 1 and n and/or 2 and n-1.
This means that the total amount of favourable cases (elements of the first group) are a quarter of the total, hence, the probability of the event is 1/4.
28) ANSWER: x=5
Because 5*5=25.
29) ANSWER: x=4
3x^2=48
Divide both sides by 3.
x^2=16
x=4
Because 4*4=16.
-1x-2y=13
x-2y=13
-x=13+2y
x=−2y−13
*Hint: The Law of Sines is Sin A/a = Sin B/b = Sin C/c
In order to solve this equation, you will have to use this equation:
A= sin^(-1)[a sinB/b]
A= sin^ (-1) [12 (sin 46°) / 11]
A = sin^ (-1) [8.632077604/11]
A = sin ^(-1) [0.7847343276]
A = 51.69611349
Therefore, Sine A would be about 52°
The area of a trapezoid = (a + b)/2 x h
Where a and b are the lengths of the parallel sides and h is the perpendicular distance between them.
When (a + b)/2 x 6 = 30
then (a + b)/2 = 5
a + b = 10
Possible values for (a,b) are (4,6) (3,7) (2,8) (1,9) assuming a < b but (7,3) is also valid.
Any positive real value is possible so (4.8, 5.2) and (0.3, 9.7) are valid
In general {a,b both real: 0
