Independent Events

<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.&nbsp;</p> <p>In most cases, ETS will tell us if events are &quot;independent.&quot; 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.&nbsp; But ETS probably wouldn&#39;t do that. The test designers would say that these two events are &quot;independent,&quot; 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 &quot;independent&quot; 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&#39;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&#39;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 &quot;tails is due.&quot; Both lines of thinking are flawed. The probability remains $50\%$ for heads and tails.&nbsp;</p> <p>This flawed reasoning can cost you a lot of money. If you&#39;re in a casino, and you win $10$,$000$, you might think that you are &quot;hot&quot; or &quot;on fire&quot; and continue betting, assuming you cannot lose. But, as so many have discovered, there is no such thing as &quot;being hot&quot; in gambling. That&#39;s why, over a long enough period of time, the house always wins.&nbsp;</p>