Question

a) At a party, seven people meet to tell each other on the weekday they
has their birthday. Make reasonable assumptions and calculate the probability that everyone
have different weekday as birth day. 
b) Select an integer between 0 and 999 at random with equal probability for each number.
What is the probability that the number contains the number 4 at least once? 
c) A random variable X is binomial distributed, X ∼ Bin (2, p). We know that P (X = 2) =
1/16. Which would have to have the parameter? Calculate the distribution function FX
and sketch in a figure what it looks like.

Answer 1

A) No. of different weeks in a year = 365/7 = 63 (the remaining 3 days will have atleast one week day and thus will be in different week )

P(7 people having their birthdays in 7 different weekdays ) = 1/53 * 1/52 *1/51 * 1/50 * 1/49 * 1/48 *1/47 = 1.2871 * 10^(-12)

assumption - each week having a birthday is equally likely despite being any number of days

B)  Probability(the number contains the number 4 at least once) = 1 - P(number has 0 4s in it )

Probability(the number contains the number 4 at least once) = 1 - 9/10 * 9/10 *9/10 = 0.271

C) P(X= x) = nCx * p^x * (1-p)^(n-x)

P(X= 2) = 2C2 p^2 (1-p)^0 = 1/16

1* p^2 *1 = 1/16

p = 1/4 = .25 = probability of success

1-p = 1-.25 = .75 = probability of failure

x	P(X=x)	P(X<= x)
0	0.5625	0.5625
1	0.3750	0.9375
2	0.0625	1.0000