# College Statistics

Michael Feathers (@mfeathers) posted a link on twitter to a nice little statistics assignment.

If you would like to solve it yourselves, stop reading when you get to the mathematics, otherwise it will spoil your fun.

Enjoy 🙂

Currently a college senior, Jeremy has had a secret crush on Emma ever since the third grade. Two weeks ago, fearing that his feelings would forever go unrequited, he broke his silence and sent Emma a letter through Campus Mail, acknowledging his twelve/year secret romance.Now, fourteen agonizing days later, he has yet to receive a response. Hoping against hope, Jeremy and his fragile psyche are clinging to the possibility that someone’s letter was lost in the mail.

Assuming that:

1. Emma (who is actually secretly dating Jeremy’s father) has a 70% change of mailing a response if, in face, she had received the letter and
2. The Campus Post Office has a one in fifty change of losing any particular piece of mail,

what is the probability that Emma never received Jeremy’s confession of the heart?

Let $B$ represent the event that Jeremy did not receive a response; let $A_1$ and $A_2$ denote the events that Emma did and did not, respectively, receive, Jeremy’s letter. The objective is to find $P(A_2 | B)$.

Stop reading now, if you do not wish to see the proof. Not that it is very complicated, mind you, but try to extract that old knowledge from years back when you had statistics yourself.

From what we know about Emma’s behavior and the incompetence of the Campus Post Office,

$\displaystyle{P(A_1) = \frac{49}{50}, P(A_2) = \frac{1}{50}, P(B | A_2) = 1}$

Also,

$\displaystyle{P(B | A1) = P(\mbox{Jeremy receive no response} | \mbox{Emma received Jeremy's letter})}$

$\displaystyle{ = P[\mbox{Emma does not respond} \cup (\mbox{Emma responds} \cap \mbox{Post Office loses letter})]}$

$\displaystyle{ = P(\mbox{Emma does not respond}) + P(\mbox{letter is lost} | \mbox{Emma responds}) \times P(\mbox{Emma responds})}$

$\displaystyle{ = 0.30 + 0.70\frac{1}{50}}$

$\displaystyle{ = 0.314}$

Therefore,

$\displaystyle{ P(A_2 | B) = \frac{1\frac{1}{50}}{0.324\frac{49}{50} + 1\frac{1}{50}}}$

$\displaystyle{ = 0.061}$

Sadly, the magnitude of $P(A_2 | B)$ is not good news for Jeremy. If $P(A_2 | B) = 0.061$, it follows that Emma’s probability of having received the letter but not caring enough to respond was almost $94\%$. “Faint heart ne’er won fair lady,” but Jeremy would probaly be well/advised to direct his romantic intentions elsewhere.