We calculate this as 25 chance of flipping two heads

This preview shows page 2 - 5 out of 6 pages.

We calculate this as 25% chance of flipping two heads on flips one and two. Same thing for flips three and four. Let’s now call the first event p (F1&2) and the second one p (F3&4). Next, build the table . Notice that all the % contributions have to add up to 1. Table of contributions to total probability: P % contrib. p (F1&2) .25 .50 p (F3&4) .25 .50 Using the law of total probability : Total probability for two heads then two heads is (.50 x .25) + (.50 x .25) = 25%. Q3: What is the probability that you will get two heads on flips three and four, conditional on getting two heads on flips one and two? A3: Probability of two heads on p (F3&4) conditional on two heads for p(F1&2). Using Bayes Theorem, we determine that it is not very useful in this form. Here, we are looking for the probability of the second item, p (F3&4), given the first, p (F1&2). The vertical bar means “given.” Which still is not very helpful. We did not calculate it this way. We did not calculate p (F1&2|F3&4). If we had, we would know the answer is just 1 minus the result. But we did calculate each part’s relation to the whole. So use the form of Bayes’ theorem that uses the law of total probability as the denominator : Which is p (F3&4)’s proportion to the total probability. From the table we made, we already know all of those pieces on the right-hand side: Example Problem Interpretation: The conditional probability that you will flip two heads in a row after previously flipping two heads in a row is 50%. Bear in mind that the raw probability of flipping two
heads in a row in independent trials is 25% and flipping four heads in a row in independent trials is 6.25%.
Recap of the Method Used in Example Problem: 1. Define success of an individual trial. 2. Use the binomial equation to calculate the cumulative probability of each part of the system. 3. Make a table with each part’s p and its percentage contribution to the whole. The percentages must add to 1. 4. Calculate the p (Total) using the law of total probability. 5. Use the total probability form of Bayes’ theorem to calculate the conditional probability for the item of interest.

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture