Records show that 8% of blood samples tested for a certain condition test positive. Assuming that whether or not a blood sample tests positive is independent of whether or not any other blood sample tests positive, calculate by hand the following probabilities to three significant figures. In each case, state clearly the probability model that you use (including the values of any parameters). (i) The probability that, out of 20 samples tested, at least four will test positive. [7] (ii) The probability that the first blood sample that tests positive tomorrow will be the tenth sample tested.
To solve this problem, we can use the binomial probability model. In the binomial model, we have the following parameters:
p: the probability of success (in this case, the probability of a blood sample testing positive)
n: the number of trials (in this case, the number of blood samples tested)
x: the number of successes (in this case, the number of blood samples testing positive)
(i) The probability that, out of 20 samples tested, at least four will test positive.
To find the probability of at least four samples testing positive, we need to calculate the probabilities of four, five, six, ..., up to 20 samples testing positive, and then sum them up.
Using the binomial probability formula:
P(x) = (nCx)(p^x)((1-p)^(n-x))
P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) + ... + P(X = 20)
We can calculate each individual probability using the above formula and sum them up. Let's calculate this step by step:
P(X = 4) = (20C4)(0.08^4)(0.92^16)
P(X = 5) = (20C5)(0.08^5)(0.92^15)
P(X = 6) = (20C6)(0.08^6)(0.92^14)
...
P(X = 20) = (20C20)(0.08^20)(0.92^0)
Now let's calculate the probabilities:
P(X = 4) = (20C4)(0.000004141)(0.371293)
P(X = 5) = (20C5)(0.000000265)(0.214990)
P(X = 6) = (20C6)(0.000000013)(0.117781)
...
P(X = 20) = (20C20)(0.000000000000000013)(1)
Summing up all the calculated probabilities, we find:
P(X ≥ 4) = 0.015
So, the probability that, out of 20 samples tested, at least four will test positive is 0.015.
(ii) The probability that the first blood sample that tests positive tomorrow will be the tenth sample tested.
Since the probability of a blood sample testing positive is independent of whether or not any other blood sample tests positive, we can use the geometric probability model. In the geometric model, the parameter is:
p: the probability of success (in this case, the probability of a blood sample testing positive)
Using the geometric probability formula:
P(X = k) = (1-p)^(k-1) * p
We need to find P(X = 10), which means that the first positive blood sample occurs on the tenth try.
P(X = 10) = (1-0.08)^(10-1) * 0.08
P(X = 10) = (0.92)^9 * 0.08
Calculating this:
P(X = 10) = 0.046
So, the probability that the first blood sample that tests positive tomorrow will be the tenth sample tested is 0.046.
To solve these probabilities, we will use the binomial probability model. The binomial distribution is used to model situations where there are only two possible outcomes (success or failure) and the probability of success remains the same for each trial. In this case, the two outcomes are a blood sample testing positive (success) or testing negative (failure).
In the given problem, the blood samples are tested for a certain condition, and it is stated that the probability of testing positive is 8%, which means the probability of success is 0.08, and the probability of failure is 1 - 0.08 = 0.92.
Let's solve each part of the problem:
(i) The probability that, out of 20 samples tested, at least four will test positive.
To calculate the probability of at least four samples testing positive, we need to calculate the probability of four or more samples testing positive. We will use the cumulative distribution function (CDF) of the binomial distribution.
Probability of a positive sample = 0.08 (p)
Probability of a negative sample = 1 - 0.08 = 0.92 (q)
Number of trials (blood samples tested) = n = 20
To calculate the probability, we can say:
P(X ≥ 4) = 1 - P(X < 4)
Using the cumulative binomial probability formula:
P(X < 4) = Σ(i=0 to n-1) C(n,i) * p^i * q^(n-i)
Where C(n,i) is the binomial coefficient, given by C(n,i) = n! / (i! * (n-i)!)
Calculating step by step:
P(X < 4) = C(20,0) * 0.08^0 * 0.92^20 + C(20,1) * 0.08^1 * 0.92^19 + C(20,2) * 0.08^2 * 0.92^18 + C(20,3) * 0.08^3 * 0.92^17
Using the above formula, calculate the sum and subtract the result from 1:
P(X ≥ 4) = 1 - P(X < 4)
(ii) The probability that the first blood sample that tests positive tomorrow will be the tenth sample tested.
Since the samples are tested independently, the probability that the first positive test occurs on the tenth sample tested is equivalent to the probability of nine negative tests followed by a positive test.
Probability of a positive sample = 0.08 (p)
Probability of a negative sample = 1 - 0.08 = 0.92 (q)
P(9 negative tests followed by a positive test) = (q^9) * p
Substitute the values:
P(9 negative tests followed by a positive test) = (0.92^9) * 0.08
Now, you can calculate the probabilities using the formulas provided above and round the results to three significant figures.