# Segment 5. Bernoulli Trials

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### Problems

#### To Compute

1. You throw a pair of fair dice 10 times and, each time, you record the total number of spots. When you are done, what is the probability that exactly 5 of the 10 recorded totals are prime?

2. If you flip a fair coin one billion times, what is the probability that the number of heads is between 500010000 and 500020000, inclusive? (Give answer to 4 significant figures.)

#### To Think About

1. Suppose that the assumption of independence (the first "i" in "i.i.d.") were violated. Specifically suppose that, after the first Bernoulli trial, every trial has a probability Q of simply reproducing the immediately previous outcome, and a probability (1-Q) of being an independent trial. How would you compute the probability of getting n events in N trials if the probability of each event (when it is independent) is p?

2. Try the Mathematica calculation on slide 5 without the magical "GenerateConditions -> False". Why is the output different?

### Class Activity

http://projecteuler.net/problem=267

**Part 2:** The problem as stated multiplies your wager by 2 on a win. What is the smallest this factor can be while still leaving the probability of ending up above one billion greater than 0.5, assuming that you play with an optimal f given the value of the factor?