twenty percent of drivers drining between 11pm and 3am are drunken drivers. using the binomial probabilities table. fine the problability that in a random sample of 12 drivers drining between 11 pm and 3am , two to four will be drunken drivers
I don't have a binomial probabilities table, so I've done the math from scratch: you should be able to verify the probabilities against your tables as you go:
The probability of a driver being drunk is 0.2, so the probability of a driver NOT being drunk is 1 - 0.2 = 0.8. So the probability of all 12 drivers not being drunk is (0.8)^12 = 0.0687.
The probability of exactly one being drunk is 12*(0.2)*((0.8)^11) (because there are 12 different ways in which one driver out of the 12 can be drunk).
The probability of exactly two being drunk is (12*11/2)*((0.2)^2)*((0.8)^10) (because there are 12*11/2 ways in which this can happen).
The probability of exactly three being drunk is (12*11*10/(3*2))*((0.2)^3)*((0.8)^9) (because there are 12*11*10/(3*2) ways in which this can happen).
The probability of exactly four being drunk is (12*11*10*9/(4*3*2))*((0.2)^4)*((0.8)^8) (because there are 12*11*10/(3*2) ways in which this can happen).
In general, the probability of there being K drunk drivers out of a total of N is:
Fact(N)/(Fact(K)*Fact(N-K))
*((0.2)^K)*((0.8)^(N-K))
where Fact(K) means K*(K-1)*(K-2)*...*1, so you should be able to check that the above formula gives the same answers for 0, 1, 2, 3 and 4 drunks as written above.
You can work out the probability of there being exactly 5, 6, ..., 12 drunk drivers in your sample by working out all the remaining terms in this sequence (and you ought really to do that, just to check that the total probability of all the possible outcomes equals 1.0), but you now have all the information you need to answer the question.
To find the probability that two to four drivers out of a random sample of 12 driving between 11 pm and 3 am will be drunk, we can use the binomial probability formula:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes
- n is the sample size (12 drivers in this case)
- k is the number of successes (2, 3, or 4 in this case)
- p is the probability of success (20% or 0.2 in this case)
- (n C k) denotes the binomial coefficient, which calculates the number of ways to choose k successes out of n trials
To use the binomial probabilities table for this calculation, follow these steps:
1. Determine the values of p and n:
- p = 0.2 (probability of a driver being drunk)
- n = 12 (sample size)
2. Identify the range of k values:
- We need to find the probability for k = 2 to 4 (inclusive)
3. Calculate the probability for each k value:
- For k = 2:
P(X = 2) = (12 C 2) * (0.2^2) * (0.8^10)
- For k = 3:
P(X = 3) = (12 C 3) * (0.2^3) * (0.8^9)
- For k = 4:
P(X = 4) = (12 C 4) * (0.2^4) * (0.8^8)
4. Add up the probabilities for all k values:
- P(2 ≤ X ≤ 4) = P(X = 2) + P(X = 3) + P(X = 4)
You can consult a binomial probabilities table to find the corresponding values of (n C k) or use a calculator or software that has built-in binomial probability functions.
However, here's a specific calculation using the binomial coefficient formula for each k value:
P(X = 2) = (12 C 2) * (0.2^2) * (0.8^10)
= (12! / (2! * (12-2)!)) * 0.04 * 0.1073741824
= 66 * 0.04 * 0.1073741824
≈ 0.2834678413
P(X = 3) = (12 C 3) * (0.2^3) * (0.8^9)
= (12! / (3! * (12-3)!)) * 0.008 * 0.134217728
= 220 * 0.008 * 0.134217728
≈ 0.2210542167
P(X = 4) = (12 C 4) * (0.2^4) * (0.8^8)
= (12! / (4! * (12-4)!)) * 0.0016 * 0.16777216
= 495 * 0.0016 * 0.16777216
≈ 0.118964953
Finally, add up the probabilities:
P(2 ≤ X ≤ 4) ≈ P(X = 2) + P(X = 3) + P(X = 4)
≈ 0.2834678413 + 0.2210542167 + 0.118964953
≈ 0.623486010