In a certain city district, the need for money to buy drugs is given as the reason for 65% of all thefts. What is the probability that exactly 2 of the next 4 theft cases reported in this district resulted from the need for money to buy drugs?
To find the probability of exactly 2 out of the next 4 theft cases resulting from the need for money to buy drugs, you can use the binomial probability formula. The formula is:
P(X=k) = (nCk) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of exactly k successes,
- nCk is the number of ways to choose k items from a set of n items (combination),
- p is the probability of success in a single trial,
- (1-p) is the probability of failure in a single trial,
- k is the number of successes,
- n is the total number of trials.
In this case,
- k = 2, as we want exactly 2 theft cases resulting from the need for money to buy drugs,
- n = 4, as there are 4 theft cases being reported,
- p = 0.65, as the probability of a theft resulting from the need for money to buy drugs is given as 65% (0.65).
Now, let's calculate the probability:
P(X=2) = (4C2) * 0.65^2 * (1-0.65)^(4-2)
To calculate 4C2 (4 choose 2):
4C2 = (4!)/(2!(4-2)!)
= (4 * 3 * 2!)/(2! * 2!)
= (4 * 3)/2
= 6
P(X=2) = 6 * 0.65^2 * 0.35^2
= 6 * 0.4225 * 0.1225
≈ 0.396
Therefore, the probability that exactly 2 out of the next 4 theft cases reported in this district resulted from the need for money to buy drugs is approximately 0.396, or 39.6%.