Q.4.23 - A First Course in Probability

Question

Consider a random collection of $n$ individuals. In approximating the probability that no $3$ of these individuals share the same birthday, a better Poisson approximation than that obtained in the text (at least for values of $n$ between $80$ and $90$ ) is obtained by letting $E_i$ be the event that there are at least 3 birthdays on day$i, i = 1, ... , 365.$

(a) Find $P\left(E_i\right)$.

(b) Give an approximation for the probability that no$3$individuals share the same birthday.

(c) Evaluate the preceding when $n=88$ (which can be shown to be the smallest value of$n$for which the probability exceeds.$5$).

Step-by-Step Solution

Answer

(a)$P\left(E_i\right)=1-e^{-n/365}-\frac{n}{365}{e}^{-n/365}-\frac{(n/365{)}^2}{2}{e}^{-n/365}$

(b)$P(Y=0)=e^{-365p}$

(c)$P(Y=0)\approx 0.5$

Step 1 : Given information(part a)

Given in the question that consider a random collection of $n$ individuals. In approximating the probability that no $3$ of these individuals share the same birthday, a better Poisson approximation than that obtained in the text (at least for values of $n$ between 80 and 90 ) is obtained by letting $E_i$ be the event that there are at least 3 birthdays on day $i,i=1,\dots ,365$.

Step 2 : Explanation

Define random variable $X$ that marks the number of people that have birthday on $i th$ day. Hence, the event $E_i$ is equivalent to the event $X\ge 3$. We have that

$P\left(E_i\right)=P(X\ge 3)=1-P(X=0)-P(X=1)-P(X=2)$

Observe that $X$ can be approximated with Poisson distribution with parameter $\lambda =\frac{n}{365}$. Hence

$P(X=0)=e^{-n/365},P(X=1)=\frac{n}{365}{e}^{-n/365},P(X=2)=\frac{(n/365{)}^2}{2}{e}^{-n/365}$

thus,

$P\left(E_i\right)=1-e^{-n/365}-\frac{n}{365}{e}^{-n/365}-\frac{(n/365{)}^2}{2}{e}^{-n/365}$

Step 3 : Final answer

$P\left(E_i\right)=1-e^{-n/365}-\frac{n}{365}{e}^{-n/365}-\frac{(n/365{)}^2}{2}{e}^{-n/365}$

Step 4 : Given information(part b)

Given in the question that,consider a random collection of n individuals. In approximating the probability that no $3$of these individuals share the same birthday, a better Poisson approximation than that obtained in the text (at least for values of n between $80$ and $90$) is obtained by letting $Ei$be the event that there are at least $3$ birthdays on day$i, i = 1, ... , 365.$

Step 5 : Explanation

Define $p$ as the number calculated in (a). Define$Y$as the random variable that counts the number of days that have the property that at least three people have birthday on that day. Hence, $Y~Binom(365,p)$, which can be approximated with Pois $(365 p)$. The event that no three people share birthday is equivalent to the event $Y=0$. Hence

$P(Y=0)=e^{-365p}$

Step 6 : Final answer

$P(Y=0)=e^{-365p}$

Step 7 : Given information(part c)

Given in the question that,consider a random collection of $n$ individuals. In approximating the probability that no$3$ of these individuals share the same birthday, a better Poisson approximation than that obtained in the text (at least for values of $n$ between$80$ and$90$) is obtained by letting $Ei$ be the event that there are at least $3$ birthdays on day $i, i = 1, ... , 365$. 

Step 8 : Explanation

We are required to calculate $P(Y=0)$ for $n=88$. Let's find $p$ for that $n$. We have that

$p=1-e^{-88/365}-\frac{88}{365}{e}^{-88/365}-\frac{(88/365{)}^2}{2}{e}^{-88/365}=1.9515·{10}^{-}3$

So we have 

$P(Y=0)=e^{-365p}=e^{-0.7122965}\approx 0.5$

Step 9 : Final answer

$P(Y=0)\approx 0.5$