For a group of 50 people, assuming that each person is equally likely to have a birthday on each of 365 days in the year, compute
(a) The expected number of days of the year that are birthdays of exactly 3 people:

0.0177 days

Step-by-step explanation:

Solution:-

- The group has n = 50 people

- Each person is equally likely to have a birthday on each of 365 days in the year.

- The probability of success p = 1 / 365

- The probability of failure q = (1 - 1/365)

- We will denote random variable X as a given day has k number of birthdays.

- X follows binomial distribution:

                            X ~ Bi ( 50 , 1/365)

- The probability that there would be 3 birthdays on a day would be:

                           P ( X = 3 ) = 50C3 ( 1 / 365)^3 * ( 1 - 1/365)^47

                           P ( X = 3 ) = 0.00035

- The expected number of days for exactly three people to have same birth date:

                          E( X = 3 ) = P (X=3)*n = 0.00035*50 = 0.0177 days

a) one student is chosen from classroom a, then a student is chosen from classroom b.  

step-by-step explanation:

independent events are ones in which the probability of one event does not affect the probability of the second event.

one student is chosen from classroom a; this does not affect the chances of choosing a student from classroom b.   this makes these independent events.

when one card is chosen from a standard deck and then set aside before a second card is chosen, the probability of the second card being chosen changes.   this makes them dependent events.

when one student is chosen from classroom a and then a second student is chosen from classroom a, the probability of choosing the second student changes.   this makes them dependent events.

when a letter is chosen from the alphabet and then a second letter is chosen, the probability of choosing the second letter changes.   this makes them dependent events.