Travis: Segment 5

From Computational Statistics (CSE383M and CS395T)
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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?

The probability of rolling a prime number with two dice is <math> \frac{15}{32} </math>.

Prime numbers in 2-dice roll
+ 1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12

The probability of getting exactly n successes in N trials is given by the probability mass function:

<math> f(n;N,p) = {N\choose n}p^n(1-p)^{N-n}</math>

So,

<math> f(5;10,\frac{15}{36}) = {10\choose 5}\bigg(\frac{15}{36}\bigg)^5\bigg(\frac{21}{36}\bigg)^{5} = 0.2138...</math>

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.)