Both words refer to fulfilling a person’s needs, wants, demands, or desires. Satisfy, which comes from a Latin word meaning “to do enough,” means to do or provide enough to fulfill a desire or meet a need or requirement (satisfied his wish for amusement, satisfies the math requirements). Someone who has been satisfied is content, no longer feeling a lack. Someone who is gratified experiences pleasure along with satisfaction. Gratify can imply pleasing someone by satisfying their explicit wish or desire, but you can also gratify vanity, pride or other selfish or base desires—perhaps by providing admiration or recognition. Depending on the context, gratify can have a negative connotation of indulgence or a neutral or positive connotation (gratified to hear of her friend’s recovery).

Mislead and delude both refer to bringing about errors of thought, judgment, or conduct. Mislead suggests leading someone into thinking untrue things or doing the wrong thing, with or without an intent to deceive (misled by the decades-old profile photo, ads that mislead consumers). It stops short of outright deception or lies, but is still usually negative. Delude always implies deception. It refers to inducing beliefs in what is imaginary or unreal. Its most frequent use is self-reflexive: people deluding themselves, out of ignorance, egotism, or wishful thinking (deluding herself into believing she could sing). Typical external agents of delusion are rhetoric, images, and leaders, whose target is often the public. The deluded are often characterized as pitiable or pathetic victims of their own minds.

Both adjectives refer to things done or occurring without a definite aim, reason, or pattern. Random suggests something occurring by chance or accident (random thoughts, random acts of violence, a random assortment). A random event has as much chance of happening as any other thing like it. There is no more reason for something arbitrary than there is for something random, but arbitrary implies something done by whim or personal preference rather than according to some discernible logic or rationale (listed in an arbitrary order, an arbitrary decision, an arbitrary deadline). For this reason, arbitrary can suggest unreasonableness, irrationality, and even injustice. These connotations, however, come more from the other, earlier definitions of arbitrary, which refer to a decision made by an individual judge or arbiter rather than by following the law.