1 Ch:5 (Ex:7)

A die is rolled until the first time T that a six turns up

1.1 What is the probability distribution for T?

Chance of getting 6 while rolling a dice is \(\frac {1}{6}\)

Now use formula: \(P(T = n) = q^{n-1}\times p\) where \(q = 1-p\)

\(P(n) = q^{n-1}\times p\)

\(P(n) = (1-\frac{1}{6})^{n-1}\times\frac{1}{6}\)

\(P(n) = (\frac{5}{6})^{n-1}\times\frac{1}{6}\)

curve((1/6)*((5/6)^(x-1)), from = 0, to=50)

1.2 Find P(T>3)

\(P(T > k) = q^{k}\)

\(P(T > 3) = (1-\frac{1}{6})^{3}\)

\(P(T > 3) = {(\frac{5}{6})}^{3} = \frac {125}{216}\)

1.3 Find P(T>6|T>3)

Let’s use text book formula \(P(T>r+s|T>s) = q^{s}\)

\(P(T>6|T>3) = (1-p)^{3}\)

\(P(T>6|T>3) = (1-\frac{1}{6})^{3}\)

\(P(T>6|T>3) = {(\frac{5}{6})}^{3} = \frac {125}{216}\)