According to a recent studey, 1 in every 4 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled twenty-five women and asked each whether she has been a victim of domestic avbuse at some point in her life. Fidn dthe probavility that at least 2 of the women sampled have been the victim of domestic abuse.
Well, if we want to find the probability that at least 2 of the women sampled have been victims of domestic abuse, we can actually calculate the probability that 0 or 1 of them have been victims and subtract that from 1.
Let's work it out. The probability that one specific woman has been a victim of domestic abuse is 1/4, which means the probability that she hasn't been a victim is 3/4. Since the sampling is independent, the probability that all 25 women have not been victims is (3/4)^25.
Now, let's calculate the probability that exactly one woman has been a victim. We can choose one woman out of the 25 in (25 choose 1) ways, and since the probability that one specific woman has been a victim is 1/4, we have (25 choose 1) * (1/4) * (3/4)^24.
Finally, we can calculate the probability that 0 or 1 woman has been a victim by adding the probabilities we just calculated:
P(0 or 1 woman has been a victim) = (3/4)^25 + (25 choose 1) * (1/4) * (3/4)^24
And now, to get the probability that at least 2 women have been victims, we subtract this from 1:
P(at least 2 women have been victims) = 1 - P(0 or 1 woman has been a victim)
Now let's just crunch the numbers... but don't worry, I won't crunch them too loudly, we don't want to disturb anyone!
*humming circus tune while calculating*
To find the probability that at least 2 of the women sampled have been the victims of domestic abuse, we need to use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
where:
- P(X = k) is the probability of getting exactly k successes (in this case, k women being victims of domestic abuse)
- n is the number of trials (25 women sampled)
- k is the number of successes (2 or more women being victims of domestic abuse)
- p is the probability of success on each trial (1 in 4 women, or 0.25)
- (n C k) represents the number of combinations of n items taken k at a time
To find the probability of at least 2 women being victims of domestic abuse, we need to calculate the probability of getting exactly 2, exactly 3, and so on, up to 25 (since we can have any number from 2 to 25).
P(at least 2 women being victims of domestic abuse) = P(X = 2) + P(X = 3) + ... + P(X = 25)
Let's calculate this step-by-step:
Step 1: Calculate P(X = k) for each value of k from 2 to 25.
P(X = k) = (25 C k) * (0.25)^k * (1 - 0.25)^(25 - k)
Step 2: Sum up all the probabilities:
P(at least 2 women being victims of domestic abuse) = P(X = 2) + P(X = 3) + ... + P(X = 25)
If you provide me with the values of (25 C k) for each k, I can help you calculate the probabilities.
To find the probability that at least 2 of the women sampled have been the victim of domestic abuse, we need to use the concept of binomial probability. The binomial distribution is used to model situations where there are only two possible outcomes (success or failure) and each trial is independent.
In this case, the probability of success is the proportion of women who have been victims of domestic abuse, which is stated as 1 in every 4. Therefore, the probability of success (p) is 1/4, and the probability of failure (q) is 3/4.
Now, let's calculate the probability using the binomial distribution formula:
P(X ≥ 2) = 1 - P(X < 2)
To find P(X < 2), we need to calculate the probabilities of having 0 and 1 women who have been victims of domestic abuse, and then subtract that from 1.
P(X < 2) = P(X = 0) + P(X = 1)
P(X = 0) = C(n, 0) * (p^0) * (q^(n-0))
= C(25, 0) * (1/4)^0 * (3/4)^(25-0)
Using the binomial coefficient formula: C(n, k) = n! / (k! * (n-k)!)
where n is the number of trials and k is the number of successes.
C(25, 0) = 25! / (0! * (25-0)!)
= 1
P(X = 1) = C(n, 1) * (p^1) * (q^(n-1))
= C(25, 1) * (1/4)^1 * (3/4)^(25-1)
Using the binomial coefficient formula:
C(25, 1) = 25! / (1! * (25-1)!)
= 25! / (1! * 24!)
= 25
Now, substitute these values back into the equation:
P(X < 2) = 1/4^0 * 3/4^25 + 25 * (1/4)^1 * (3/4)^(25-1)
Evaluate P(X < 2) using a calculator:
P(X < 2) ≈ 0.128
Finally, find P(X ≥ 2) by subtracting P(X < 2) from 1:
P(X ≥ 2) = 1 - 0.128
P(X ≥ 2) ≈ 0.872
Therefore, the probability that at least 2 of the 25 women sampled have been victims of domestic abuse is approximately 0.872 or 87.2%.