For this problem please REWRITE the SOLUTION in a different way.

Events

Underlying the computations of probability is an organized system for describing and working with the outcomes of chance experiments. These outcomes can be divided into two types: (1) simple events, which are the individual outcomes of an experiment and, more generally, (2) events, which consist of collections of simple events. For instance, the chance experiment of conducting a series of stress tests on three metal parts has the eight possible outcomes PPP, PPF, PFP, FPP, PFF, FPF, FFP, and FFF, where P and F denote the test results "pass" and "fail," and the order in which the letters appear corresponds to the part number tested (e.g., PPF indicates that the first two parts passed the test, but the third part failed). Each of these eight outcomes is a simple event, which, taken together, form the sample space of the experiment.

Events are often denoted by single uppercase letters, usually from the beginning of the alphabet, much like we denote constants in formulas by lowercase letters. Single-letter names for events are very useful when applying the probability formulas in Section 5.2. Thus, we might denote the event that at least two parts pass the stress test by A, the event that exactly 1 part passes the stress test by B, and so forth. Events can also be described by just listing, in brackets, the simple events that comprise them. For example, the event that at least two parts pass the stress test corresponds to the set of outcomes {PPP, PPF, PFP, FPP}. If we had also chosen to denote this event by the letter A, then we could also write A = {PPP, PPF, PFP, FPP}.

Let's continue with our example of stress-testing metal parts. Suppose that we now select and test four parts. Using sequences of Ps (for parts that pass the test) and Fs (for parts that fail the test), the sample space of the experiment of selecting and testing four metal parts is somewhat larger than that of the experiment of selecting and testing three metal parts, discussed previously. In particular, the sample space consists of these 16 simple events: {PPPP, PPPF, PPFP, PFPP, FPPP, PPFF, PFPF, PFFP, FPPF, FPFP, FFPP, PFFF, FPFF, FFPF, FFFP, FFFF}. For convenience, these events are listed in order of decreasing numbers of Ps in each four-letter sequence.

Suppose we are interested in the events A = at least two parts pass the stress test and B = at most two parts pass the stress test. In terms of simple events, we can write A and B as

A = {PPPP, PPPF, PPFP, PFPP, FPPP, PPFF, PFPF, PFFP, FPPF, FPFP,FFPP}

B = {PPFF, PFPF, PFFP, FPPF, FPFP, FFPP, PFFF, FPFF, FFPF, FFFP, FFFF}

Note that A and B have several simple events in common (shown underlined).

PLEASE REWRITE the above SOLUTION in a different way. (i.e., a way that was not written in the original answer).