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
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?
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.
To answer these questions, we will use the binomial probability formula. The formula is:
P(X = k) = (n choose k) * p^k * q^(n-k)
Where:
- P(X=k) is the probability of getting exactly k successes
- n is the number of trials (in this case, the number of murders)
- k is the number of successes (In this case, the number of victims who knew their murderer)
- p is the probability of success on each trial (In this case, the probability that a victim knew their murderer, which is 0.64)
- q is the probability of failure on each trial (In this case, the probability that a victim did not know their murderer, which is 1 - 0.64 = 0.36)
A) To find the probability that at least 35 of the victims knew their murderers, we need to add up the probabilities of getting 35, 36, 37, ..., 63 successes.
P(X >= 35) = P(X = 35) + P(X = 36) + ... + P(X = 63)
Using the binomial probability formula, we can calculate each term individually and then add them up.
B) To find the probability that at most 48 of the victims knew their murderers, we need to sum up the probabilities of getting 0, 1, 2, ..., 48 successes.
P(X <= 48) = P(X = 0) + P(X = 1) + ... + P(X = 48)
C) To find the probability that less than 30 victims did not know their murderer, we need to sum up the probabilities of getting 0, 1, 2, ..., 29 failures (which is the complement of knowing the murderer).
P(X < 30) = P(X = 0) + P(X = 1) + ... + P(X = 29)
D) To find the probability that more than 20 victims did not know their murderer, we need to add up the probabilities of getting 21, 22, ..., 63 failures.
P(X > 20) = P(X = 21) + P(X = 22) + ... + P(X = 63)
Let's calculate the probabilities step-by-step.
To calculate the probabilities, we can use the binomial distribution formula. The binomial distribution is used when there are two possible outcomes (success or failure) for each trial, and the trials are independent of each other.
In this case, the probability of a victim knowing their murderer is 64%, which means the probability of a victim not knowing their murderer is 36% (100% - 64%).
Let's calculate the probabilities for each scenario:
A) At least 35 victims knew their murderers:
We need to calculate the probability of having 35, 36, ..., 63 victims who knew their murderers. To do this, we can calculate the probability of each scenario individually and add them up.
P(at least 35 victims knowing their murderers) = P(35) + P(36) + ... + P(63)
The probability of having exactly k victims who knew their murderers (where k is between 35 and 63) is given by the binomial distribution formula:
P(k) = (nCk) * (p^k) * (q^(n-k))
Where n is the number of trials (63 unsolved murders), p is the probability of success (0.64), q is the probability of failure (0.36), and (nCk) represents the number of ways to choose k objects out of n.
Using this formula, we can calculate the probability of each scenario and add them together to find the answer for A.
B) At most 48 victims knew their murderers:
We can use the complement rule to calculate this probability. The complement rule states that P(A) = 1 - P(not A).
P(at most 48 victims knowing their murderers) = 1 - P(49) - P(50) - ... - P(63)
C) Less than 30 victims did not know their murderer:
Similar to scenario A, we need to calculate the probability of having 0, 1, ..., 29 victims who did not know their murderer, and add them up.
P(less than 30 victims not knowing their murderer) = P(0) + P(1) + ... + P(29)
D) More than 20 victims did not know their murderer:
Again, we can use the complement rule to calculate this probability.
P(more than 20 victims not knowing their murderer) = 1 - P(0) - P(1) - ... - P(20)
These calculations can be done using a spreadsheet software or programming language with the necessary functions to calculate binomial probabilities.