Question

(a) make a two - way table that describes the sample space in terms of events b and e.
(b) find ( p(b) ). (round to 2 decimal places. leave your answer in decimal form.)
find ( p(e) ). (round to 3 decimal places. leave your answer in decimal form.)
(c) find the probability of getting a blue 8. (round to 3 decimal places.) (leave your answer in decimal form)
(d) explain why ( p(bcup e)
eq p(b)+p(e) ).
the events \disk is blue\ and \disk is the number eight\ these two events occur at the same time.
use the general addition rule to compute ( p(bcup e) ). (round to 3 decimal places.) (leave your answer in decimal form)

Explanation:

Step1: Calculate \(P(B)\)

The formula for probability is \(P(B)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
From the table, the total number of outcomes \(n = 36\), and the number of outcomes for event \(B\) (total of column \(B\)) \(n(B)=4\).
So, \(P(B)=\frac{4}{36}\approx0.11\)

Step2: Calculate \(P(E)\)

The total number of outcomes \(n = 36\). The number of outcomes for event \(E\) (total of row \(8\)): from the table, if we assume the total of non - 8 is \(32\), then \(n(E)=4\) (since \(36 - 32=4\)).
\(P(E)=\frac{4}{36}\approx0.111\)

Step3: Calculate the probability of getting a blue 8

The number of blue 8s (intersection of blue and 8) is \(n(B\cap E) = 1\). Using the formula \(P(B\cap E)=\frac{n(B\cap E)}{n}\), with \(n = 36\), we get \(P(B\cap E)=\frac{1}{36}\approx0.028\)

Step4: Explain \(P(B\cup E)

eq P(B)+P(E)\)
The general addition rule is \(P(B\cup E)=P(B)+P(E)-P(B\cap E)\). Since \(P(B\cap E)
eq0\) (the events “disk is blue” and “disk is the number eight” are not mutually - exclusive, they can occur at the same time), \(P(B\cup E)
eq P(B)+P(E)\)

Step5: Calculate \(P(B\cup E)\)

Using the general addition rule \(P(B\cup E)=P(B)+P(E)-P(B\cap E)\)
We know \(P(B)=\frac{4}{36}\), \(P(E)=\frac{4}{36}\), \(P(B\cap E)=\frac{1}{36}\)
\(P(B\cup E)=\frac{4 + 4-1}{36}=\frac{7}{36}\approx0.194\)

Answer:

(b) \(P(B)\approx0.11\)
\(P(E)\approx0.111\)
(c) \(\approx0.028\)
(d) The events are not mutually - exclusive (they can occur simultaneously)
\(P(B\cup E)\approx0.194\)