1

A = Selecting red ball
p(A) = 54 / (54 + 9 + 75)
B = Selecting blue ball
p(B) = 75 / (54 + 9 + 75)

p(A or B) = P(A) + p(B) = 54 / (54 + 9 + 75) + 75 / (54 + 9 + 75) = .9348

2

A = Selecting red golf ball
p(A) = 20 / (19 + 20 + 24 + 17) = 0.25

3

p(Male or Doesn't live with parents) = 1 - p(Male AND live with parents) = 1 - 215/1399 = 0.8463

4

They are most likey to be Dependent events. This can be confirmed only with additional information.

5

Since the veggie wraps needs to be different, this can be treated as a arrangement problem
ie. 8 * 7 * 6 * 7 * 6 * 5 * 3 = 211,680

6

They are Independent events.

7

First spot can be filled by any of 14 eligible members, next by 13 and so on till 8th spot.
ie  
(14 * 13 * 12 * 11 * 10 * 9 * 8 * 7)
## [1] 121080960

8

Number of ways of selecting 0 red jelly beans = 9C0
Number of ways of selecting 1 orange jelly beans = 4C1
Number of ways of selecting 0 green jelly beans = 9C3

Total number of ways of selecting 3 jelly beans = 22C4

Total probability = (9C0 + 4C1 + 9C3) / 22C4 = 0.0459

9

11 * 10 * 9 * 8
## [1] 7920

10

The complement is 33% of subscribers to a fitness magazine are 34 years old or younger.

11

Using a binomial random variable formula, we get the probability of getting exactly 3 heads as follows.
 => NCx*p^x*(1-p)^(N-x) = C(4,3) * 0.5^3 * 0.5^1 = 0.25
 
 p(win) = 0.25
 p(loss) = 1 - p(win) = 0.75 
 
 Step 1: E[proposition] = 97 * 0.25 - 30 * 0.75 = $1.75
 Step 2: E[559 times step 1] = 559 * 1.75 = $978.25

12

Using a binomial random variable formula, => NCx*p^x*(1-p)^(N-x)

Probability of 0 tails = 9C0 * .5^0 * (1 - .5)^9 = 1 * .5^9
Probability of 1 tails = 9C1 * .5^1 * (1 - .5)^8 = 9 * .5^9
Probability of 2 tails = 9C2 * .5^2 * (1 - .5)^7 = 36 * .5^9
Probability of 3 tails = 9C3 * .5^3 * (1 - .5)^6 = 84 * .5^9
Probability of 4 tails = 9C4 * .5^4 * (1 - .5)^5 = 126 * .5^9

p(4 tail or less) = (1 + 9 + 36 + 84 + 126) * (1/2)^9 = 256/512 = 0.5
p(win) = 0.5
p(loss) = 1 - 0.5 = 0.5

Step 1 : E[X] = 23 * 0.5 - 0.5 * 26 = $-1.50
You will loose $1.50
Step 2 : 499 * -1.50 = -1491
You will loose $1491

13

table

table

Based on the tree diagram above, following questions are solved.
a.
0.2 * 0.59 / (0.2 * 0.59 + 0.8 * 0.1)
## [1] 0.5959596
b.
0.8 * 0.9 / (0.8 * 0.9 + 0.2 * 0.41)
## [1] 0.8977556
c.
P(AUB) = P(A) + P(B) - P(A AND B)
A = Randomly selected individual is a liar
B = Randomly selected individual identified as liar by polygraph

P(A) = 0.2
P(B) = 0.2 * 0.59 + 0.8 * 0.1 = 0.198
P(A AND B) = 0.2 * 0.59 = 0.118
P(A OR B) = 0.2 + 0.198 - 0.118 = 0.28