Question

We roll a fair die four times.

(a) Describe the sample space Ohm and the probability measure P that models this experiment (i.e., give the value P(omega) for each outcome omega elementof Ohm).

(b) Let A be the event that there are at least two fives among the four rolls. Let B be the event that there is at most one five among the four rolls. Find the probabilities P(A) and P(B), by finding the ratio of the number of favorable outcomes to the total, as in the Equally Likely Outcomes rule.

(c) What is the set AUB? What equality should P(A) and P(B) satisfy? Check that your answers to part (b) satisfy this equality.

Answer 1

Answer:

A and B are complementary events

P(A)+P(B) =1

Step-by-step explanation:

a) sample space will consist of 6x6x6x6 events as

(1,1,1,1) to (6,6,6,6)

P(each event) = $\frac{1}{6^4}$

b) A- there are atleast two 5 among the four rolls.

B = atmost one 5 among four rolls

Let X be the no of 5's in the four rolls. X is binomial since each die is independent of the other with p = 1/6 and n =4

P(A) = $P(X\geq 2)\\=\Sigma _2^5 (5Cr)\frac{1}{6} (\frac{5}{6} )^{4-r}$

=0.131944

P(B) = $P(X\leq 1) = 0.868094$

We find that A and B are complementary events.

c) P(AUB) =1 since A and B are complementary