1.) Assume 365 days in the year and that people\'s birthdays are randomly distri

Question

1.) Assume 365 days in the year and that people's birthdays are randomly distributed throughout the year. With 16 people in the room, what is the probability that at least 2 have the same birthday? How many people are required so that the probability is at least 50%?

2.) A factory has 3 machines (A, B,&C)

A makes 20% of the parts produced by the factory, B makes 30% of the parts and C makes 50% of the parts.

6% of the parts made by A are defective, 7% of the parts made by B are defective and 8% of the parts made by C are defective.

Al the parts are tossed in a single box. What is the probability that a part picked from the box is defective?

Suppose that a part is selected from the box at day's end and is found to be defective. What is the probability that C made it? That B made it? That A made it?

Explanation / Answer

Let us assume that all 365 possible birthdays are equally likely.

Let there are n people at a party.

P(atleast 2 people share their b'day) = 1-P(none of the guests share their b'day)

Choose any n days out of the 365 day calendar year and assign 1 person to each day.

Thus the number of ways of doing this is (365 P n).

The total number of ways of assigning birthdays to n people without any restrictions is 365*365*365...n times= 365^n
Thus

P(none of the guests share their b'day) = (365 P n/365^n)

n=16

P(atleast 2 people share their b'day) =1-(365 P 16/365^16) = 0.284

Thus we want

(365 P n/365^n) < 0.5

which can be numerically found out to be satisfied for the first time for n = 23.

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