What are the chances that a person who is murdered actually knew the murderer? About 64% of people who are murdered actually knew the murderer. Suppose that a detective file in New Orleans has 63 unsolved murders. What is the probability that

Question

A )at least 35 of the victims knew their murderers?
B) at most 48 of the victims knew their murderers?
c)less than 30 victims did not know their murderer?
d) More than 20 victims did not know their murderer?

Answer 1

To solve these types of problems, we can use the binomial probability formula:

P(X = k) = C(n, k) * p^k * (1-p)^(n-k)

Where:
- P(X = k) is the probability of having exactly k successes in n trials
- C(n, k) is the number of combinations of n items taken k at a time, which can be calculated using the formula: C(n, k) = n! / (k!(n-k)!)
- p is the probability of success (in this case, the probability that a victim knew their murderer, which is 64% or 0.64)
- n is the number of trials (in this case, the number of unsolved murders, which is 63)
- k is the number of successful outcomes (for example, the number of victims who knew their murderers)

To calculate the probabilities for each of the scenarios given, we can use the given formula and sum the probabilities for the given range of k.

A) At least 35 of the victims knew their murderers:
We need to find the probability P(X >= 35). The sum of probabilities of k from 35 to 63:

P(X >= 35) = SUM [P(X = k) for k = 35 to 63]

B) At most 48 of the victims knew their murderers:
We need to find the probability P(X <= 48). The sum of probabilities of k from 0 to 48:

P(X <= 48) = SUM [P(X = k) for k = 0 to 48]

C) Less than 30 victims did not know their murderer:
Since less than 30 victims did not know their murderer, this means that more than 33 victims (63 - 30) knew their murderer. We need to find the probability P(X > 33). The sum of probabilities of k from 34 to 63:

P(X > 33) = SUM [P(X = k) for k = 34 to 63]

D) More than 20 victims did not know their murderer:
Since more than 20 victims did not know their murderer, this means that fewer than 43 victims (63 - 20) knew their murderer. We need to find the probability P(X < 43). The sum of probabilities of k from 0 to 42:

P(X < 43) = SUM [P(X = k) for k = 0 to 42]

Note that calculating these probabilities manually is tedious and time-consuming. You can use statistical software or a calculator with binomial probability functions to find these probabilities more easily.