<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-independent-events" target="_blank">Independent Events Exercise</a></p><p><strong><span style="color:#8e44ad;">Independent Events</span></strong> are happenings or occurrences that <strong><u>DO NOT</u></strong> influence each other. For example, does one coin flip influence the second? Does one roll of a die influence the second? What about if we flip a coin and get ten heads in a row? Does this influence the 11th flip? NOPE. The probability of getting heads on that 11th filp is still $\frac{1}{2}$, the same as all other flips of the coin. </p>
<p>In most cases, ETS will tell us if events are "independent." For example, they <strong><u>would not</u></strong> say this:</p>
<p style="margin-left: 40px;"><em>The probability that Sarah graduates high school is $90\%$ and the probability that Bill passes is $80\%$</em>.</p>
<p>Do these events influence each other? Perhaps. Maybe Bill sees Sarah studying hard and also begins studying harder. But ETS probably wouldn't do that. The test designers would say that these two events are "independent," which thankfully makes the math easier.</p>
<p><strong><span style="color:#2980b9;">Two Independent Events Can Happen Simultaneously</span></strong></p>
<p>One might erroneously assume that if two events are "independent" that they cannot happen at the same time. This is not accurate. If two events are independent, they <u>can happen simultaneously</u>. For example, imagine the probability that Bob drinks a soda at lunch is $0.40$ and the probability that Bill drinks a soda at lunch is $0.65$. And Bob and Bill don't know each other. In fact, they live in different states. These are independent events. Is it possible for Bob and Bill to BOTH drink a soda at lunch? Sure!</p>
<p><strong><span style="color:#27ae60;">Some Common Examples of Independent Events</span></strong></p>
<ul>
<li>the flipping of a coin twice or more</li>
<li>the flipping of multiple coins</li>
<li>drawing a card out of a deck, putting it back, and then drawing again</li>
<li>spinning a spinner twice or more</li>
<li>weather on consecutive days</li>
<li>random generation of multiple numbers</li>
</ul>
<p><strong><span style="color:#2980b9;">The Gambler's Fallacy</span></strong></p>
<p>The human brain is a flawed thing. For many people, getting $10$ consecutive heads means the eleventh roll is likely also heads. Or perhaps a person takes the opposite tack and assumes that "tails is due." Both lines of thinking are flawed. The probability remains $50\%$ for heads and tails. </p>
<p>This flawed reasoning can cost you a lot of money. If you're in a casino, and you win $10$,$000$, you might think that you are "hot" or "on fire" and continue betting, assuming you cannot lose. But, as so many have discovered, there is no such thing as "being hot" in gambling. That's why, over a long enough period of time, the house always wins. </p>