CPhill

I know the first one  :

P (3 men )   =   C(3,3)  / P (5,3)  =  1 / 10

P (two tails )   =  C(7,2) (3/4)^2(1/4)^5  =    189/16384  ≈ 1.15 %

B / Y  =  8/5   ⇒  B  = (8/5)Y

So

[ (8/5)Y - 12]  / [ Y + 21 ]  =  1/3      cross-multiply

(8/5)Y - 12  =  (1/3) [ Y + 21]

(8/5)Y - 12  = (1/3)Y + 7       

(8/5)Y - (1/3)Y  =  7 + 12

[24 - 5 ] Y / 15  =  19     ultiply both sides by  15

19Y  =  285       divide both sides by 19

Y  =  15 

B = (8/5)(15) =  24 

P ( L l R )   =  P(L and R)  / P (R)

Rearranging this  we have that

P(R)  =  P (L and R)  / P(L l R)  = 0.03  / 0.0625 

 Second one.....we have  60 girls....18 prefer a clay court...so the probability of choosing a girl at random who prefers a clay court  is  18 / 60  =  3 / 10

Maybe  36 ??

1   2   3   4   6   9   12   18

3  and then an even  =  (1/8) (4/8)  = 4/64  =  1/16

Intersection of L and T  are all the elements common to both T and L

Note...there are no  common  elements....so  {empty set }

Third one...we have 13 cards

2 and 3 without replacement = (2/13) * (1/12)  = 2/156  =  1/78

2 and 5 without replacement  = (2/13) (2/12) =  4/156 = 1/39

3 and 2 with replacement  = (1/13) (2/13)  =  2/ 169

5 and 2 with replacement  =  (2/13) (2/13)  =  4/169

We have 10 tiles in total....6 even, 4 odd

Picking two even  numbered tiles without replacement  = 6/10 * 5/9 = 30/90 = 1/3

Picking an even  numbered tile and then an odd number tile  with replacement  = 

6/10  *  4/10  =  24/100  =  6/25

Picking two  odd numbered tiles with replacement = 4/10 * 4/10  = 16/100 =  4/25

Picking an odd and then an even  without replacement  = 4/10 * 6/9  = 24/90 = 4/15

The first has the greatest probability

"C"  is the correct answer for the second

I assume that this might be :

5

Σ  [ 5(-2)^n-1   =       55

n = 1

\(\log_{4}3=x$. Then $\log_{2}27=kx\)       find k

In exponential form we have.....

4^x  = 3           and        2^(kx) = 27

Working with the expression on  the left :

4^x  = 3      and we can write

(2^2)^x  = 3

2^(2x)  = 3

Cube each side

[ 2^(2x)]^3  = 3^3

2^(6x)  =  27

So  k  = 6

Ths base is opposite the 30° angle

The height is opposite the 60°  angle  and  is √3 as long as the base =  5√3  units

So.....the area  = (1/2) base * height  =

(1/2) (5) *(5√3)  =

12.5√3  units^2   ≈  21.7 units^2