A Magic Coin

He found Emma in the library after school at their usual table. As he approached, that infectious grin of hers spread across her face and eventually willed his own mouth to reflect the gesture. She was wearing a short, pink skirt which barely covered her shapely rear end and a buttoned top that he suspected did not have a bra underneath. Her attire probably broke at least seven rules within the school's dress code, but no teacher would dream of punishing a girl as sweet as Emma, so for her the code was more of a guideline.

"I've got a surprise for you Isaac," she said clapping her hands together excitedly, "I need you to pick a number, one or two."

Inwardly he groaned, "Emma, please, can I just have one normal day of you being my girlfriend? I'm sorry after everything, it's just too much, all at once. If we're going to be together, could we just start with one normal day, just us?"

Emma's quickly transformed into one of disappointment, her smile dissolved by his words, but was quickly hid by a smaller smile than before.

"Okay then, just boyfriend and girlfriend."

It was nice to finally sit down with Emma again after school, just the two of them, without all the sexual tension.

Emma was having difficulty understanding conditional probability, which, as Mr. Poole had emphasised, was sure to be a major part of the upcoming exam.

Despite Isaac's best efforts, each time he tried to explain her face still became a portrait of incomprehension, her beautiful features scrunched up in incomprehension like a paper ball.

"I just don't get it. What's the difference between one event happening and then another, and the intersection between two events?"

Isaac did his best not to show signs of despair, he knew that what came easy to him was not the same for Emma, but it felt as if he had been explaining this concept over and over for the last hour.

"It's not that conditional probability measures the probability of one event occurring after another, it is the probability that one event has occurred given another has occurred. Like, if I roll a dice twice, the first roll won't affect the outcomes of the second roll since the probability of every face of the dice is the same regardless of how many times I roll it, so in that case intersection would be the same as conditional probability, but if I had five marbles in a bag and ---"

Isaac cut himself off, as he realised that once again, he had lost Emma. He would have found her furrowed face cute if not for the fact that it was this expression that had been the signal for dead end after dead end for all his attempts to get her to grasp conditional probability. He knew she was trying her best; a change of tactics was needed, maybe if she saw it applied in a real question, he could get it across?

"Okay, here's the worksheet Poole gave to my class a few weeks ago, read this question and then we'll go through it."

She nodded and turned towards the piece of paper. Isaac smiled as he saw her squidge her nose slightly, it reminded him of a mouse sniffing a piece of cheese. She did it every time she was concentrating while reading, and it was so endearing that it unravelled the noose of annoyance that had characterised the library session thus far. She was his girlfriend, his little mouse and he loved her, no matter how terrible she was at math.

A magic coin changes its odds each time its flipped. On the first flip, the probability of a heads and tails are equal. However, on the second, if the first flip resulted in heads then there is a three-tenths probability that it will result in heads again, and if the first flip resulted in tails then heads has an equal change. What is the probability that given the second flip was heads, the first flip was also heads?

Christ, he'd forgotten how terrible Poole could be at writing problems. A good teacher, but there was a reason he taught math and not literature. Really? A magic coin? There's no way he could go through this without confusing Emma even more, but he'd committed to it, so he might as well try.

"Alright, uhm... So if I flip a coin like this," Isaac mumbled as he pulled a coin out of his wallet and threw it upwards, "then the question wants us to believe that if I flipped it aga---"

"Wait!", barked Emma, "what face did the coin land on?"

"What?"

"The one you just threw, how did it land?"

"Emma that's not really important, I just ---"

"Please Isaac," her eyes were twinkling, but her voice was filled with a **** urge, an irresistible pleading that defied rationality, dissolving resistance like salt in water.

Looking at the palm of his hand, Isaac stated the result of the toss, resigned that this was yet another failed explanation.