Robert came home from school, tired and upset. All he really wanted to do was sleep, so sleep he did. "The whole game was rigged," he muttered before falling asleep.

He wakes up in a familiar setting. Soon, the Number Devil walks into his classroom.

"Rough day?" the Number Devil asks. "Just another rigged raffle. Those are impossible anyway," Robert responded.

"I wouldn't be so sure, Robert. Back then, I was fond of Statistics," the Number Devil replied.

"Yes, you dimwit, statistics. Like how you're wrong 95% of the time, or how you irritate me with your lack of mathematical knowledge 85% of the time. Those statistics!" said the Number Devil as his voice booms with anger. Robert replies, "well, while you're riled up, I suppose you wouldn't bother teaching me?" The number Devil replies, "Gladly."

"You see, Robert, mathematicians do more than just find patterns in hopping numbers, prima-donnas, rutabagas, and all that. In math heaven, if my memory serves me well, loved to delve into applicable methods of math as much as they loved swimming in concepts."

And so, Robert told the Number Devil about the first raffle scenario earlier in school.

"My teacher placed 3 objects, one gold and the other two were bronze, behind three doors. Mr. Bockel told me to choose a door, so I picked door one. Afterwards, he opened another door with a bronze item in it. He asked if I wanted to switch doors. I didn't. When I opened my door, it was bronze item."

Robert complained, and complained about how it was all a trick. After a while, the Number Devil chimes in "Or is it?" He then proceeded to wave his cane, and they appear on a game show.

"Now let me explain why it would've been better to switch doors. First, some context. This problem was loosely based on the American game show Let's make a Deal named after its original host, Monty Hall. It became famous when Marilyn vos Savant quoted it on a column in a magazine. This problem became known to be the Monty Hall Problem. Let's use two rabbits and a car."

"Please, no more rabbits." Robert whines.

"Fine, I'll be keeping the car."

The Number Devil asked Robert to pick a door. Like earlier, he picked door one.

"Well, what are the chances that the door you picked has the car?" asks the Number Devil.

"1 out of 3?"

"That's indeed correct."

The Number Devil opens door two, which reveals a goat.

"Now, what are the chances that the door you picked does not have the car, knowing that door two has a goat?"

"Also 1 out of 3."

The Number Devil waves his cane, and a table with the possible outcomes appeared.

"As you can see, if you chose to switch, you would've had a 2/3 chance to win the car, compared to a 1/3 chance if you didn't."

"And it would've worked with even fifty doors!"

It began to make sense to Robert as he nodded in agreement.

"So, the better strategy would have been to switch doors every time." Robert added.

"So, what about the next event after lunch?"

"Mr. Bockel wanted us to guess the probability that at least two people shared a birthday in our class. Of course, I believed that would have been impossible!"

The Number Devil laughed, snickered, and chuckled to tears.

"Haven't I taught you that anything is possible? But, perhaps your negativity would help us with this problem," said the Number Devil as he wiped tears from his eyes.

"Now, what makes you say that?" Robert asks.

The Number Devil scratches his head and replies, "Well, we have 365 possible birthdays, and 23 students in your class. The probability of at least two people NOT sharing birthdays is:"

"And the probability that at least two people out of three don't share birthdays is:"

"So if we jump ahead, the probability that at least two people out of 23 don't share birthdays is:"

Robert, now amazed, shouts "That just means that the probability of two out of two people sharing birthdays is .28%, the probability of 2 out of 3 is .83%, and the probability of 2 out of 23 is 50.7%! That makes no sense!"

"It's no fun when you get optimistic. The math just adds up that way. Nevertheless, yes. The math checks out, your brain just doesn't. You'd be surprised how high the hopping numbers hop!" exclaims the Number Devil.