Question

question 1
a 6 - sided die rolled twice. let e be the event \the first roll is a 5\ and f the event \the second roll is a 5\.
(a) are the events e and f independent?
input yes or no:
(b) find the probability of showing a 5 on both rolls. write your answer as a reduced fraction.
answer:

Explanation:

Step1: Determine independence

Two events \(E\) and \(F\) are independent if \(P(E\cap F)=P(E)\times P(F)\).
The probability of rolling a \(5\) on a single - roll of a 6 - sided die is \(P(E) = P(F)=\frac{1}{6}\).
The probability of \(E\cap F\) (first roll is \(5\) and second roll is \(5\)): The total number of outcomes when rolling a die twice is \(n(S)=6\times6 = 36\). The favorable outcome for \(E\cap F\) is \((5,5)\), so \(n(E\cap F) = 1\) and \(P(E\cap F)=\frac{1}{36}\).
Since \(P(E)\times P(F)=\frac{1}{6}\times\frac{1}{6}=\frac{1}{36}=P(E\cap F)\), the events \(E\) and \(F\) are independent.

Step2: Calculate \(P(E\cap F)\)

As mentioned above, using the formula for the probability of independent events \(P(A\cap B)=P(A)\times P(B)\) (where \(A = E\) and \(B = F\)).
We know \(P(E)=\frac{1}{6}\) and \(P(F)=\frac{1}{6}\).
So \(P(E\cap F)=\frac{1}{6}\times\frac{1}{6}\).

Answer:

(a) Yes
(b) \(\frac{1}{36}\)