Statistics show that about 5% of all males are colorblind. Suppose that 20 males are selected at random.
What is the probability that at least 4 of the 20 people are colorblind?
My answer: I got 1-0.984
The answer key says 0.00257394
at least 4 means
4 or 5 or 6 or ... or 20 are colourblind
so exclude cases of none, 1 , 2, and 3
colourblind = .05
not colourblind = .95
None:
C(20,0) (.05)^0 (.95)^20 = .358486
one:
C(20,1) (.05) (.95)^19 = .37735..
two:
C(20,2) (.05)^2 (.95)^18 = .1886768
three:
C(20,3)(.05)^3 (.95)^17 = .059582..
adding up those 4 and subtracting from 1 I got .0159
which is the same result as you had
mmmhhhh?
To calculate the probability that at least 4 out of 20 males are colorblind, we can use the binomial probability formula.
First, we need to find the probability of selecting a colorblind male. Since 5% of all males are colorblind, the probability of selecting a colorblind male is 0.05.
Next, we use the binomial probability formula, considering that we want to find the probability of at least 4 colorblind males out of 20. The formula is as follows:
P(X >= k) = 1 - P(X < k-1)
In this case, k is 4. Therefore, we need to calculate P(X < 4).
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
To calculate each term, we use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
where n is the number of trials (20), k is the number of successful trials (colorblind males), and p is the probability of success (0.05).
P(X = 0) = (20 choose 0) * (0.05)^0 * (1 - 0.05)^(20 - 0)
P(X = 1) = (20 choose 1) * (0.05)^1 * (1 - 0.05)^(20 - 1)
P(X = 2) = (20 choose 2) * (0.05)^2 * (1 - 0.05)^(20 - 2)
P(X = 3) = (20 choose 3) * (0.05)^3 * (1 - 0.05)^(20 - 3)
Once we calculate those values, we can sum them to find P(X < 4). Finally, to find the probability of at least 4 colorblind men, we subtract P(X < 4) from 1.
Using a calculator or statistical software, the answer is approximately 0.00257394, which matches the answer key you provided.