There are 30 people in a room. Assume 365 days in the year, and that each person

Question

There are 30 people in a room. Assume 365 days in the year, and that each person is equally likely to be born on any of the days. What is the probability that

a) no two people have the same birthday?

b) exactly three people have the same birthday, while any others are born on different days?

c) Suppose you walk into a room with 30 people. What is the probability that at least one person has the same birthday as you?

d) what is the probability that exactly two people have the same birthday as you?

Explanation / Answer

Answer to the question)
n = 30 people

.

Part a)

Total number of options = (365)^30

Number of ways that 30 people have different date of birth = 365*364*...*336

Thus Probability that no two persons have same birthdate = 365*364*...336 /365^30

Probability = 0.29368

.

Part b)

If exactly 3 persons have same date of birth then probability = (365/365)*(1/365)*(1/365)

The probability of rest of the 27 people is : (364*363*,,,338) / (364)^27

total probability = (364*363*...338)/(365^2 * 364^27)

.

Part c)

Probability that atleast 1 has same date = 1 - probability that none has same date as you

Probability none has same date = (365*364...335) / (365)^31 = 0.2695

Probability atleast one has same date = 1 - 0.2695 = 0.7305

.

Answer to part d)

Probability that exactly two persons have date of birth as mine = 1/365^2

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