A box contains 10 balls. Three of them are white, five are red, and two are

black. Calculate the following probabilities:
a) The first ball extracted is red.
b) The first ball extracted is red or white.
c) The second ball extracted is red given that the first is white (with replacement).
d) The second ball extracted is red given that the first is white (without replacement).
e) The second ball extracted is red given that the first is also red (without
replacement).
f) How many samples of three balls can be extracted from this population? Assume that you are extracting one sample of size three

A box contains 10 balls. Three of them are white, five are red, and two are

black. Calculate the following probabilities:
a) The first ball extracted is red.
b) The first ball extracted is red or white.
c) The second ball extracted is red given that the first is white (with replacement).
d) The second ball extracted is red given that the first is white (without replacement).
e) The second ball extracted is red given that the first is also red (without
replacement).
f) How many samples of three balls can be extracted from this population? Assume that you are extracting one sample of size three

a) prob(red) = 5/10 = 1/2

b) prob(red or white) = 7/10
c) prob( as stated) = 5/10 or 1/2 , there are still 10 balls, since you replaced it whatever happened in first draw has no effect on the second draw
d) prob( as stated) = 5/9 , now it matters, there are only 9 balls left
e) 4 reds left, 9 in total, so prob = 4/9

f)
We can list them, there are only a few

WWW
RRR
BBB
WWR
WWB
RRW
RRB
BBW
BBR
WRB

To calculate the probabilities, we need to know the number of favorable outcomes and the total number of possible outcomes. Let's calculate each probability step by step:

a) The first ball extracted is red.
- Favorable outcomes: There are 5 red balls.
- Total outcomes: There are 10 balls in total.
- Probability = Favorable outcomes / Total outcomes = 5/10 = 1/2

b) The first ball extracted is red or white.
- Favorable outcomes: There are 5 red balls and 3 white balls.
- Total outcomes: There are 10 balls in total.
- Probability = Favorable outcomes / Total outcomes = (5+3)/10 = 8/10 = 4/5

c) The second ball extracted is red given that the first is white (with replacement).
- In this case, after extracting the first white ball, the box still contains 10 balls, with 5 red balls.
- The probability remains the same as in part a: 1/2.

d) The second ball extracted is red given that the first is white (without replacement).
- After removing the first white ball, the total number of balls becomes 9 (since there are only 9 balls left in the box).
- Favorable outcomes: There are still 5 red balls remaining.
- Total outcomes: There are 9 balls in total.
- Probability = Favorable outcomes / Total outcomes = 5/9

e) The second ball extracted is red given that the first is also red (without replacement).
- After removing the first red ball, the total number of balls becomes 9.
- Favorable outcomes: There are 4 red balls remaining.
- Total outcomes: There are 9 balls in total.
- Probability = Favorable outcomes / Total outcomes = 4/9

f) To calculate the number of samples of three balls that can be extracted from this population, we can use the concept of combinations.
- The total number of balls is 10, and we want to extract three balls, so we can calculate the number of combinations of 10 balls taken 3 at a time.
- The formula for combinations is nCr = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items to choose.
- In this case, n = 10 and r = 3.
- Number of samples = 10! / (3! * (10-3)!) = 120

So, there are 120 possible samples of three balls that can be extracted from this population.

I hope this explanation helps! Let me know if you have any further questions.