Slack is a partnership between two traders: Simon and Jack.

Simon and Jack each invested money in different commodity futures. Simon’s coffee trades can lose 3.5million Euros with probability 1/3 or earn a profit of 1 million Euros with probability 2/3. Jacks’s cereal trades can lose 1 million Euros with probability 1/4 or earn a profit of 1 million Euros with probability 3/4.

1. What is the probability that Slack’s profits will be non-negative

(larger than or equal to 0)?

2. What are Slack’s expected profits (in millions)?

3. If the coffee and cereal trades are statistically independent, what is

the probability that Slack’s profits will be exactly equal to zero?

Answer:

The sample space of Slack's trades, denoted by ?S,? is given as

?S={Loss for Simon and loss for Jack,Loss for Simon and profit for Jack,Profit for Simon and loss for Jack,Profit for Simon and profit for Jack}?

Case 1: Loss for Simon and loss for Jack

Net profit ?=−3.5 m−1 m=−4.5 m? (Negative sign means Slack incurred a loss of 4.5 million Euros)

Probability ?=(1/3)(1/4)=1/12?

Case 2: Loss for Simon and profit for Jack

Net profit ?=−3.5 m+1 m=−2.5 m?

Probability ?=(1/3)(3/4)=1/4?

Case 3: Profit for Simon and loss for Jack

Net profit ?=1 m−1 m=0 m?

Probability ?=(2/3)(1/4)=1/6?

Case 4: Profit for Simon and profit for Jack

Net profit ?=1 m+1 m=2 m?

Probability ?=(2/3)(3/4)=1/2?

Now, the net profit earned by slack, given in the sample space is

?S={−4.5 m,−2.5 m,0 m,2 m}?.

And, the respective probabilities are

p={1/12,1/4,1/6,1/2}?

Answer (1):

Slack's profits will be non-negative for two cases, one in which Simon earns profit and Jack incurs loss, and the second case in which both earn profit.

Hence, required probability ?=(1/6)+(1/2)=2/3?

So, Slack's profits will be non-negative with a probability of ?2/3.?

Answer (2):

Slack's expected profits is given as

?E(S)? ?=i=1∑4?pi?Si??

??E(S)=(1/12)(−4.5 m)+(1/4)(−2.5 m)+(1/6)(0 m)+(1/2)(2 m)?

??E(S)=0 m?

So, the expected profits of Slack are ?0? million Euros.

Answer (3):

Slack's profits are exactly equal to 0 when Simon earns profit and Jack incurs loss.

So, required probability ?=(1/6)?

Hence, the probability that Slack's profits are 0 is equal to ?(1/6).?