Question

the accompanying venn diagram describes the sample - space of a particular experiment and events a and b. complete parts a and b below.
a. suppose the sample points are equally likely. find p(a) and p(b).
p(a)=0.4
p(b)=0.2
(round to two decimal places as needed.)
b. suppose p(1)=p(2)=p(4)=p(5)=p(8)=\frac{1}{20} and p(3)=p(6)=p(7)=p(9)=p(10)=\frac{3}{20}. find p(a) and p(b).
p(a)=
p(b)=
(round to two decimal places as needed.)

Explanation:

Step1: Identify sample - points in A

Count sample - points in A from Venn - diagram. A has 4 sample - points: 3, 4, 7, 9.

Step2: Calculate P(A) in part a

Since sample points are equally likely and total number of sample points is 10, and number of sample points in A is 4. So $P(A)=\frac{4}{10}=0.4$.

Step3: Identify sample - points in B

Count sample - points in B from Venn - diagram. B has 3 sample - points: 4, 5, 6.

Step4: Calculate P(B) in part a

Since total number of sample points is 10, and number of sample points in B is 3. So $P(B)=\frac{3}{10}=0.3$.

Step5: Calculate P(A) in part b

$P(A)=P(3)+P(4)+P(7)+P(9)$. Given $P(3) = P(4)=P(7)=P(9)=\frac{3}{20}$. So $P(A)=4\times\frac{3}{20}=\frac{12}{20}=0.6$.

Step6: Calculate P(B) in part b

$P(B)=P(4)+P(5)$. Given $P(4)=\frac{3}{20}$ and $P(5)=\frac{1}{20}$. So $P(B)=\frac{3 + 1}{20}=\frac{4}{20}=0.2$.

Answer:

a. $P(A)=0.4$, $P(B)=0.3$
b. $P(A)=0.6$, $P(B)=0.2$