Probability of Independent Events
Each NFL game starts with a coin toss. Heading into Super Bowl 51 (2017), the team from one of the conferences (NFC) had won \(17\) of the last \(19\) coin tosses. That is pretty impressive considering a coin toss gives you a \(50\%\) chance of winning. Does the chance of a NFC team winning the coin toss one year affect their chance of winning it the next year? Although this particular example might cause you to think otherwise, the answer is no. Each coin toss is independent and the outcome of one does not affect the outcome of the other.
Each NFL game starts with a coin toss. Heading into Super Bowl 51 (2017), the team from one of the conferences (NFC) had won \(17\) of the last \(19\) coin tosses. That is pretty impressive considering a coin toss gives you a \(50\%\) chance of winning. Does the chance of a NFC team winning the coin toss one year affect their chance of winning it the next year? Although this particular example might cause you to think otherwise, the answer is no. Each coin toss is independent and the outcome of one does not affect the outcome of the other.
If two (or more) events are independent, then the probability that both events will happen is \(P\left(A\ \text{and}\ B\right)=P\left(A\right)\cdot P\left(B\right)\). This can be extended to any other number of independent events as well.
Example 1: You select a random card from a standard deck of \(52\) playing cards and then you flip a coin. Find the probability that you select a face card and the coin lands on tails.
Example 2: About \(35\%\) of Americans will take a family vacation this year. You randomly select \(5\) Americans.
a) What is the probability that all of them will take a family vacation? What is the probability that none of them will?
\(P\left(\text{all}\right)=\left(.35\right)^5=.0053\)
\(P\left(\text{none}\right)=\left(.65\right)^5=.116\)
b) What is the probability that at least one of them will take a family vacation? What is the probability that at least one of them will not take a family vacation?