Question

births are approximately uniformly distributed between the 52 weeks of the year. they can be said to follow a uniform distribution from 1 to 53 (a spread of 52 weeks). round answers to 4 decimal places when possible.
a. the probability that a person will be born at the exact moment that week 21 begins is p(x = 21) =
b. the probability that a person will be born between weeks 9 and 14 is p(9 < x < 14) =
c. the probability that a person will be born after week 24 is p(x > 24) =
d. p(x > 11 | x < 28) =
e. find the 82nd percentile.
hint:
written hint
helpful videos: probability +, conditional probability + conditional probability + percentiles +

Explanation:

Step1: Recall uniform - distribution properties

For a continuous uniform distribution \(X\sim U(a,b)\) where \(a = 1\) and \(b = 53\), the probability density function is \(f(x)=\frac{1}{b - a}=\frac{1}{53 - 1}=\frac{1}{52}\) for \(a\leq x\leq b\) and \(0\) otherwise. The probability of a single - point in a continuous distribution is \(0\).

Step2: Calculate \(P(9\lt x\lt14)\)

The formula for \(P(c\lt X\lt d)\) in a uniform distribution \(U(a,b)\) is \(P(c\lt X\lt d)=\frac{d - c}{b - a}\). Here, \(a = 1\), \(b = 53\), \(c = 9\), \(d = 14\), so \(P(9\lt x\lt14)=\frac{14 - 9}{53 - 1}=\frac{5}{52}\approx0.0962\).

Step3: Calculate \(P(x\gt24)\)

\(P(x\gt24)=\frac{b - 24}{b - a}\), substituting \(a = 1\) and \(b = 53\), we get \(P(x\gt24)=\frac{53 - 24}{53 - 1}=\frac{29}{52}\approx0.5577\).

Step4: Calculate \(P(x\gt11|x\lt28)\)

By the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here, \(A=\{x\gt11\}\), \(B = \{x\lt28\}\), \(A\cap B=\{11\lt x\lt28\}\). \(P(A\cap B)=\frac{28 - 11}{52}=\frac{17}{52}\), \(P(B)=\frac{28 - 1}{52}=\frac{27}{52}\), so \(P(x\gt11|x\lt28)=\frac{P(11\lt x\lt28)}{P(x\lt28)}=\frac{\frac{28 - 11}{52}}{\frac{28 - 1}{52}}=\frac{17}{27}\approx0.6296\).

Step5: Calculate the 82nd percentile

The formula for the \(k\) - th percentile \(x_k\) in a uniform distribution \(U(a,b)\) is \(x_k=a+(b - a)\frac{k}{100}\). Substituting \(a = 1\), \(b = 53\), and \(k = 82\), we get \(x_{82}=1+(53 - 1)\frac{82}{100}=1 + 52\times0.82=1+42.64 = 43.64\).

Answer:

a. \(0.0000\)
b. \(0.0962\)
c. \(0.5577\)
d. \(0.6296\)
e. \(43.6400\)