Suppose we have an urn with 10 white balls and 6 red balls, and we create an experiment by randomly selecting 6 balls from the urn.
a) What is the probability that the sample has the same number of white balls as red balls?
b) What is the probability that there are more red balls than white balls in our sample?
c) What is the probability that the sample has at least 5 white balls?
Assuming that the balls are replaced, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
a) (6/16)^3 * (10/16)^3 = ?
If they are not replaced,
a) 6/16 * 5/15 * 4/14 * 10/13 * 9/12 * 8/11 = ?
Depending on replacement or not, go through the similar processes with the remaining problems.
b) Here you are considering 4 or 5 red balls. "More" indicates that there has to be at least one white ball. (Either-or probabilities are found by adding the individual probabilities.)
c) Here you are considering 5 or 6 white balls
a). 300/1001
To solve these probability problems, we need to find the total number of possible outcomes (sample space) and the number of outcomes that fulfill the given conditions.
a) To find the probability that the sample has the same number of white balls as red balls, we need to determine the number of outcomes where we select an equal number of white and red balls.
Step 1: Find the number of ways to select 3 white balls (same as selecting 3 red balls) out of 10 white balls:
We will use the combination formula for this: C(n, r) = n! / (r! * (n - r)!)
C(10, 3) = 10! / (3! * (10 - 3)!) = 120
Step 2: Find the number of ways to select the remaining 3 balls from the remaining 6 red balls:
C(6, 3) = 6! / (3! * (6 - 3)!) = 20
Step 3: Multiply the number of ways from Step 1 and Step 2 together to get the total number of ways to select an equal number of white and red balls:
Total ways = 120 * 20 = 2400
Step 4: Find the total number of possible outcomes by selecting any 6 balls out of the 16 balls:
C(16, 6) = 16! / (6! * (16 - 6)!) = 8008
Step 5: Calculate the probability by dividing the favorable outcomes (Step 3) by the total outcomes (Step 4):
Probability = 2400 / 8008 = 0.2998 (rounded to four decimal places)
Therefore, the probability that the sample has the same number of white balls as red balls is approximately 0.2998.
b) To find the probability that there are more red balls than white balls in our sample, we need to determine the number of outcomes where we select more red balls than white balls.
Step 1: Find the number of ways to select 4 red balls out of the 6 red balls:
C(6, 4) = 6! / (4! * (6 - 4)!) = 15
Step 2: Find the number of ways to select the remaining 2 white balls from the 10 white balls:
C(10, 2) = 10! / (2! * (10 - 2)!) = 45
Step 3: Multiply the number of ways from Step 1 and Step 2 together to get the total number of ways to select more red balls than white balls:
Total ways = 15 * 45 = 675
Step 4: Calculate the probability by dividing the favorable outcomes (Step 3) by the total outcomes (Step 4):
Probability = 675 / 8008 = 0.0843 (rounded to four decimal places)
Therefore, the probability that there are more red balls than white balls in our sample is approximately 0.0843.
c) To find the probability that the sample has at least 5 white balls, we need to determine the number of outcomes where we select 5 or 6 white balls.
Step 1: Find the number of ways to select 5 white balls out of the 10 white balls:
C(10, 5) = 10! / (5! * (10 - 5)!) = 252
Step 2: Find the number of ways to select the remaining 1 red ball from the 6 red balls:
C(6, 1) = 6! / (1! * (6 - 1)!) = 6
Step 3: Find the number of ways to select all 6 white balls:
C(10, 6) = 10! / (6! * (10 - 6)!) = 210
Step 4: Add the number of ways from Step 1, Step 2, and Step 3 together to get the total number of ways to select at least 5 white balls:
Total ways = 252 + 6 + 210 = 468
Step 5: Calculate the probability by dividing the favorable outcomes (Step 4) by the total outcomes (Step 4):
Probability = 468 / 8008 = 0.0584 (rounded to four decimal places)
Therefore, the probability that the sample has at least 5 white balls is approximately 0.0584.