Segment 5 Sanmit Narvekar

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Segment 5

To Calculate

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?

We use the formula from the slides, making the appropriate substitutions (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = 10, n = 5, p = \frac{15}{36}} (15 of the possible 36 dice rolls sum to a prime number)):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{N}{n} p^n (1-p)^{N-n} = \binom{10}{5} (15/36)^5 (21/36)^5 = 0.2138 }


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.)

Here is the code:

cdfs = binocdf([500010000, 500020000], 10^9, 0.5);
cdfs(2)-cdfs(1)

The resulting answer is: 0.1606


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?

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