This question is a far deeper question than it first appears. The wording of the question suggests that random events can and do hold the power to influence our future lives. The reason that this question is deep is because it assumes that random events exist. That means philosophies such as fatalism and predestination are automatically assumed false. These would believe that seemingly random events are not random at all.
A variable in this question is what exactly the question means by "random." For example, let's say that you have to take English 2 at some point during high school. The class is taught by two different teachers, and the school registrar assigns you teacher "B." Is that random? It could be, especially if the registrar rolled a set of dice to determine your placement, such as even numbered rolls = section A and odd numbered rolls = section B.
It's possible that the above situation has a huge and important impact on your life. Teacher "B" might be so good and so inspiring about literature you decide that pursuing a career such as an English teacher, editor, or publisher is what you want to do with your life.
The previous example is more of a long-term random event. A student would interact with that teacher for months; however, a random event could be short and equally impactful. Let's say that you are out for a walk, and you witness a car accident. You immediately call 911 and rush to the cars to check on people. You feel completely helpless as to what to do. The EMTs show up soon after, and you are amazed at how calmly they assess the situation and begin administering care to some gruesome injuries. The event causes you to change your college major from business to medicine, because you want to be able to help people in need the way that you saw those men and women do it.
Anecdotally, I had a high school friend who was involved in a bad motorcycle accident. His spinal cord was severed, and he became paralyzed from the waist down. He is now a minister and inspirational speaker. If you asked him how he got into that particular line of work, his answer always involves the accident, his recovery, and the people involved with his recovery. None of those post-accident things would have happened if the random accident didn't happen.