Question

Five people have just won a $100 prize, and are deciding how to divide the $100 up between them. Assume that whole dollars are used, not cents. Also, for example, giving $50 to first person and $10 to the second is different from vice versa.

(a) How many ways are there to divide up the $100, such that each gets at least $10?

// why is it (54 choose 4) and not (50 choose 4)?

(b) Assume that the $100 is randomly divided up so that each person gets at least $10, with all of the possible allocations counted in (a) equally likely. Let X represent the amount the first person receives. State the state space SX.

(c) Find the the expected amount of money that the first person receives. You can leave your answer as a sum

can you add little footnotes/explanations? Please? Thanks so much in advance!!

Answer 1

Say a, b, c, d and e are the amount of money each person gets. Hence, we have to find the number of solutions of the following equation:

a+b+c+d+e = 100

a) If each person gets atleast $10: a,b,c,d,e>=10

Say, a1 = a - 10, b1 = b - 10 ....

Therefore, the equation becomes:

a1 + b1 + c1 + d1 + e1 = 50

The number of solutions of the above equation are: C(n+r-1, r-1)

r = 5, n = 50

Hence, the number of ways are C(54 ,4)

b) The state space of Sx will be: (10, 11, 12, 13..... 58, 59, 60)

c) Expected money can be found by the following:

Say when a1 = x (Belonging to Sx)

The number of ways are the solution to the equation:

b1 + c1 + d1 + e1 = 50 - x

Number of ways = C(54-x, 3)

Probability = C(54-x, 3)/C(54,4)

Hence,