Worksheet 6: Conditioning

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1. Take a moment to think of an example of an uncertain situation from real life, in which you learned information \(B\) that dramatically changed your estimate of whether some \(A\) was going to happen.

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The formula for the conditional probability of \(A\) given \(B\) is \(\Pr[A \mid B] = \frac{\Pr[A \cap B]}{\Pr[B]}.\)

2. Can you justify this formula? \vspace{2.5cm}

3. Suppose I flip a fair coin twice. Let \(A\) be the event that the first coinflip comes up heads, and let \(B\) be the event that at least one of the coinflips comes up heads.

   a. What is \(\Pr[A \mid B]\)? \vspace{2.5cm}

   b. What is \(\Pr[B \mid A]\)? \vspace{2.5cm}

Distracted driving. Let \(A\) be the event that you are driving distracted, and let \(B\) be the event that you get in a car accident.

According to the National Highway Traffic Safety Administration, in recent years about \(13\%\) of car accidents involve distracted driving.

4. How would you phrase this in the language of conditional probability? \vspace{2.5cm}

5. What is \(\Pr[B \mid A]\), in plain English? \vspace{2.5cm}

6. What do you think happens more frequently: distracted driving, or car accidents? \vspace{2.5cm}

7. Do you think \(\Pr[B \mid A]\) is smaller, larger, or no different than \(\Pr[A \mid B]\)? \vspace{2.5cm}

OJ Simpson.

Let \(A\) be the event that a woman is abused, let \(M\) be the event that the woman is murdered, and let \(G\) be the event that her husband is guilty of murdering her.

8. The defense argued that only \(1/2500\) abused women are murdered by their husband. How would you phrase this using conditional probabilities? \vspace{2.5cm}

9. Can you think of events \(A,B\), where conditioning on \(B\) has no impact on \(A\)? \vspace{2.5cm}