Question

We proved in class that in a group of 23 people, the probability of two people having the same birthday is 0.5073. Also, in a group of 100 people, the probability of two people having the same birthday is 0.9999998.   On late-night television’s The Tonight Show with Johnny Carson (on air during 1962-1992), Carson was discussing the birthday problem. At a certain point, he remarked to his audience of approximately 100 people “Great! There must be someone here who was born on my birthday”. Do you agree with his statement? Why? Find the probability of at least one audience member having the same birthday as Carson.

Answer 1

The question is based on the famous "birthday paradox"...

Carson remarked to audience of approximately 100 people that “Great! There must be someone here who was born on my birthday”...

Solution :- I don't agree with Carson's statement.

It can be proved with the concept of probability. In this case Carson had audience of approximately 100 people.

So it is a group of 101 people.

Let's calculate the complementary probability of no one person from the group 100 people having the same birthday as Carson.

So if we take any random person from the group , (364 / 365) is the probability of that person not having same birth date as Carson. So in the group of 100 people no one having the same birth date as Carson can be given as:-

$\small P(A') = (\frac{364}{365}) \ * ....... \ * (\frac{364}{365}) = (\frac{364}{365})^{100}$

(Event A' = no one (from the audience of 100 people) having the same birth date as Carson )

No we will calculate at least one person from the audience having same birth date as Carson ...

$\small P(A) = 1 - \ P(A') = 1 - (\frac{364}{365})^{100} \approx \textbf {0.240} \approx 24 \%$

(Event A = at least one person from the audience of 100 people is having same birth date as Carson)

So from the above Probability P(A) we can infer that it is 0.240 and not 0.9999998. And P(A) clearly shows the the statement remarked by Carson is not feasible as it carries only 24% of chance.