Conditional probability · Definition

Independence of events

probabilityconditional

Before you start you need to know

It separates phenomena that influence each other from those that do not, and it is the hypothesis that allows multiplying probabilities.

Definition

Two events \(A\) and \(B\) are independent when knowing that one occurred does not change the probability \(P\) of the other.

\[P(A\cap B)=P(A)\,P(B)\]

If \(P(B)>0\), \(A\) and \(B\) being independent is equivalent to \(P(A\mid B)=P(A)\): learning that \(B\) occurred does not change the probability of \(A\). Substituting into the product rule gives the formula \(P(A\cap B)=P(A)P(B)\), which is adopted as the definition because it also holds when a probability is zero.

Example: on a die, \(A=\) «even» and \(B=\) «1 or 2» satisfy \(P(A\cap B)=1/6=\tfrac12\cdot\tfrac13\), so they are independent even though they share the outcome 2. Independence is not decided by eye, but by checking the equality with the probabilities.

If \(A\) and \(B\) are independent, so are \(A\) and \(B^c\), \(A^c\) and \(B\), \(A^c\) and \(B^c\). For three or more events pairwise independence is not enough: the product formula must hold for every subfamily. This justifies computations such as «at least one head in three tosses» \(=1-(1/2)^3\).

Do not confuse it with being mutually exclusive: if \(A\cap B=\varnothing\) and both have positive probability, knowing that one occurs ensures the other does not, so they are dependent. The next concept, total probability, uses conditionals to combine cases.

Think about it

If \(P(A)=0.5\) and \(P(A\mid B)=0.5\) with \(P(B)>0\), what follows?

That \(A\) and \(B\) are independent: \(P(A\cap B)=P(A\mid B)P(B)=P(A)P(B)\).

Can an event be independent of itself?

Only if \(P(A)=P(A)^2\), that is, if \(P(A)\) is 0 or 1.

Exercises 10

  1. If \(A\) and \(B\) are independent with \(P(A)=0.3\) and \(P(B)=0.5\), what is \(P(A\cap B)\)?
  2. Two fair coins are tossed. What is the probability of getting two heads?
  3. If \(P(A)=0.4\), \(P(B)=0.5\) and \(P(A\cap B)=0.2\), are they independent?
  4. If \(P(A)=0.5\), \(P(B)=0.4\) and \(P(A\cap B)=0.3\), which statement is correct?
  5. If \(A\) and \(B\) are independent with \(P(A)=0.6\) and \(P(B)=0.5\), what is \(P(A\mid B)\)?
  6. If \(A\) and \(B\) are independent with \(P(A)=0.3\) and \(P(B)=0.2\), what is \(P(A\cup B)\)?
  7. A fair coin is tossed three times. What is the probability of at least one head?
  8. A fair die is rolled. Let \(A\): «even» and \(B\): «1 or 2». Are they independent?
  9. Let \(A\) and \(B\) be independent events with positive probabilities. Can they be mutually exclusive?
  10. If \(A\) and \(B\) are independent, which statement is always true?

References

  • Joseph K. Blitzstein, Jessica Hwang, Introduction to Probability, 2nd edition, CRC Press, 2019. §2.5
  • Sheldon Ross, A First Course in Probability, 10th edition, Pearson, 2019. §3.4
  • Larry Wasserman, All of Statistics, 1st edition, Springer, 2004. §1.5