Probability Question

CFA Level I
#1

In the Example Question on page 480, it asks you to find the Probability of a Limit Order Executing.

It uses two limit orders of 1.$10.00 and 2.$9.75 with probabilities of executing with an hour being 1. 0.35 and 2. 0.25. It then asks you to find the probability that either Order 1 or Order 2 will execute (i.e. addition rule for probabilities).

I successfully worked out that P(1 or 2) = 0.35 + 0.25 - 0.25 = 0.35 (Note: the P(AB) is calculated by first finding the P(AIB)m which is 1 because the Limit Order of 10.00 would have executed if the Limit Order of 9.75 has been triggered).

The bit I don’t understand is their explanation: “Note that the outcomes for which Order 2 executes are a subset of the outcomes for which Order 1 executes. After you count the the probability that Order 1 executes, you have counted the probability of the outcomes for which Order 2 also executes. Therefore, the answer to the question is the probability that Order 1 executes, 0.35.”

I don’t see how counting the probability that Order 1 executes means you have counted the probability of the outcomes for which Order 2 also executes? Would appreciate any help. Thanks.

#2

Well you have said it yourself, if the limit order of 9.75 have been executed, then you know the order of 10 would also have been executed. in other words, all the cases of limit order of 9.75 being executed are specials cases of order of 10 being executed. So the probability of 10 being executed is the same as the probability of either 10 or 9.75 being executed (cause there’s no possibility of 9.75 being executed without 10 being executed).

So you just did some calculations that were not necessary in that case, it’s not a coincidence that your probability of 2 being executed is the same number as the probability of 1 and 2 being executed (0.25).