1. The probability of getting at least one roto in 5 CDs is 0.921875 or 92.1875%.
2. The probability of getting exactly 3 rotos in 5 CDs is 0.205078 or 20.5078%.
3. The probability of getting 2 or more rotos in 5 CDs is 0.609375 or 60.9375%.
Scenario 1: Probability of getting at least one roto in 5 CDs
To calculate the probability of getting at least one roto in 5 CDs, we can use the complement rule, which states that the probability of an event A occurring is equal to 1 minus the probability of its complement (not A occurring). In this case, the event A is getting at least one roto in 5 CDs, and its complement is getting no rotos in 5 CDs.
The probability of getting no rotos in 5 CDs can be calculated using the binomial probability formula:
P(X=k) = nCk * pk * (1-p)^(n-k)
where:
X is the random variable representing the number of rotos
k is the number of successes (rotos)
n is the number of trials (CDs)
p is the probability of success (roto)
(1-p) is the probability of failure (not a roto)
In this case, k=0, n=5, and p=10/20=0.5. Substituting these values into the formula, we get:
P(X=0) = 5C0 * 0.5^0 * (1-0.5)^5 = 0.078125
Therefore, the probability of getting no rotos in 5 CDs is 0.078125.
The probability of getting at least one roto in 5 CDs is then:
P(A) = 1 - P(not A) = 1 - 0.078125 = 0.921875
Therefore, the probability of getting at least one roto in 5 CDs is 0.921875 or 92.1875%.
Scenario 2: Probability of getting exactly 3 rotos in 5 CDs
To calculate the probability of getting exactly 3 rotos in 5 CDs, we can use the binomial probability formula again. In this case, k=3, n=5, and p=10/20=0.5. Substituting these values into the formula, we get:
P(X=3) = 5C3 * 0.5^3 * (1-0.5)^2 = 0.205078
Therefore, the probability of getting exactly 3 rotos in 5 CDs is 0.205078 or 20.5078%.
Scenario 3: Probability of getting 2 or more rotos in 5 CDs
To calculate the probability of getting 2 or more rotos in 5 CDs, we can calculate the probability of getting 0 or 1 roto and subtract it from 1. The probability of getting 0 rotos is 0.078125 as calculated in Scenario 1. The probability of getting 1 roto can be calculated using the same binomial probability formula:
P(X=1) = 5C1 * 0.5^1 * (1-0.5)^4 = 0.3125
Therefore, the probability of getting 0 or 1 roto is 0.078125 + 0.3125 = 0.390625.
The probability of getting 2 or more rotos is then:
P(A) = 1 - P(not A) = 1 - 0.390625 = 0.609375
Therefore, the probability of getting 2 or more rotos in 5 CDs is 0.609375 or 60.9375%.
Complete question:
Itadori has a bag with 20 CD's, of which 10 are rotos.
Keeping in mind the following data if we extract 5 at random, calculate
Scenario 1: Probability of getting at least one roto in 5 CDs
Scenario 2: Probability of getting exactly 3 rotos in 5 CDs
Scenario 3: Probability of getting 2 or more rotos in 5 CDs