I need help with the following questions, i attempted to solve them, and i also put my answers here too.

When you are answering them, please show all your steps as i am kinda lost in this field.

1. suppose S = {1,2,3} and P({1,2}) = 1/3 and P({2,3}) = 2/3

compute P({1}), P({2}), P({3})

my answer

P({1}) + P({2}) + P({3}) = 1 <--- 1 being 100% and 0 being 0%

P({1}) + P({2}) = 1/3

P({2}) + P({3}) = 2/3

a) P({1}) + (1/3 - P({1})) + P({3}) = 1

P({3}) = 2/3

b) P({1}) + (2/3 - P({3})) + P({3}) = 1

P({1}) = 1/3

so then 1/3 + P({2}) + 2/3 = 1

so P({2}) = 0

therefore: P({1}) = 1/3, P({2}) = 0, P({2}) = 2/3

2. Suppose A1 watches the six o'clock news a 2/3 of the time and watches eleven o'clock news 1/2 of the time and watches both news 1/3 of the time. For a randomely selected day what is the probability that A1 watches only the six o'clock new?s, and whats the probability that A1 watches neither news?

my answer

chance for 6oclock = 2/3

chance for 11oclock news = 1/2 => complementary is 1/2

for only six o'clock news

has to watch the 6oclock news which is 2/3, and not watch the 11oclock news which is 1/2

2/3*1/2 = 1/3

so only watching 6 o'clock news is 1/3 chance...

for watching neither

avoid watching 6o'clock news which is 1/3 (complementary) and avoid watching eleven o'clock news 1/2 (complementary)

so its

1/3*1/2 = 1/6

so watching neither is 1/6 chance...

3. Suppose your right knee is sore 15% of the time and your left knee is sore 10% of the time. What is the largest possible percentage that at least 1 knee is sore? What is the smallest possible percentage that at least 1 knee is sore?

my answer

Largest

15%+10% = 25%

Smallest

0.15*0.10 = 0.015

so 1.5%

4. Suppose a card is randomely chosen from a standard 52 card deck.

What is the probability that the card is a jack or a club (or both) ?

my answer

chance of jack = 4/52

chance of club = 13/52

chances of club or jack = 4/52 + 13/52 = 17/52

chances of jack and club is 1/52

5. Suppose 55% of students are female and 45% are male.

44% of females have long hair and 15% of males have long hair.

What is the probability that a random student will either be female or have long hair (or both)?

my answer

chance of female: 55%

chance of long hair: (55%*44%) + (45%*15%) = 0.2420 + 0.0675 = 0.3095 = 30.95%

chance of female or long hair = 55% + 30.95% = 85.95%

chance of female with long hair: 0.2420 = 24.2%