Find the value of x in each case:

Answer:

A="b is situated in the center"

B="c lies to the right of b"

C="The letters def occur sequentially in that arrangement"

a) b can occupy 7 positions; however, only one of these is the center. Therefore, P(A)=1/7

b) Let X=i; "b holds the i-th position"

Y=j; "c occupies the j-th position"

P(B)=1/2

c) Let X=i; "d holds the i-th position"

Y=j; "e occupies the j-th position"

Let Z=k; "f is in the i-th position"

P(C)=1/42

P(A∩C)=2*(1/7*1/6*1/5*1/4)=1/420

P(B∩A)=3*(1/7*1/6)=1/14

P(A|C)=P(A∩C)/P(C)=(1/420)/(1/42)=1/10

P(B|C)=P(B∩C)/P(C)=(1/420)/(1/42)=1/10

P(A|B)=P(B∩A)/P(B)=(1/14)/(1/2)=1/7

P(A∩B)=1/14

P(A)P(B)=(1/7)*(1/2)=1/14

Events A and B are independent

P(A∩C)=1/420

P(A)P(C)=(1/7)*(1/42)=1/294

Events A and C are not independent

P(B∩C)=1/420

P(B)P(C)=(1/2)*(1/42)=1/84

Events B and C are not independent

Answer:

Step-by-step explanation:

Player A has a red marble and a blue marble, while Player B also has a red marble and a blue marble.

Therefore, the probability of selecting one marble is equal, at 0.5.

Due to the independence of A and B's choices, the joint event is calculated by multiplying the probabilities.

Let A represent the amount that player A wins.

If both players select one marble, the sample space can be considered as

             (R,R)  (R,B)  (B,R) (B,B)

Probability     0.25  0.25  0.25  0.25

A's winnings          3       -2     -2       1

E(A)      0.75   -0.5    -0.5   0.25   =    0

Thus, the game is even, offering equal expected values for both A and B.

It does not influence the outcome whether you are A or B.

Response: The properties of mathematical operations assist in solving integer-related problems as they provide various methods to arrive at the solution.

Step-by-step explanation: