randRange(7,10) randRange(3,6) randRange(2,STUDENTS-2) STUDENTS-BOYS random() < 0.5 ARE_B ? BOYS : GIRLS (factorial(NUM_B)*factorial(STUDENTS-GROUP))/(factorial(STUDENTS)*factorial(NUM_B-GROUP)) getGCD(factorial(NUM_B)*factorial(STUDENTS-GROUP),factorial(STUDENTS)*factorial(NUM_B-GROUP)) factorial(NUM_B)*factorial(STUDENTS-GROUP)/GCD factorial(STUDENTS)*factorial(NUM_B-GROUP)/GCD

There are STUDENTS students in a class: BOYS boys and GIRLS girls.

If the teacher picks a group of GROUP at random, what is the probability that everyone in the group is a ARE_B ? "boy" : "girl"?

ANSWER

One way to solve this problem is to figure out how many different groups there are of only ARE_B ? "boys" : "girls", then divide this by the total number of groups you could choose. Since every groups is chosen with equal probability, this will be the probability that a group of all ARE_B ? "boys" : "girls" is chosen.

We know two ways to count the number of groups we can choose: we use permutations if order matters, and combinations if it doesn't. Does the order the students are picked matter in this case?

It doesn't matter if we pick ARE_B ? "John" : "Julia" then ARE_B ? "Ben" : "Beatrice" or ARE_B ? "Ben" : "Beatrice" then ARE_B ? "John" : "Julia", so order must not matter. So, the number of ways to pick a group of GROUP students out of STUDENTS is \dfrac{STUDENTS!}{(STUDENTS-GROUP)!GROUP!} = \binom{STUDENTS}{GROUP}. [Show me why]

Remember that the STUDENTS! \; and (STUDENTS-GROUP)! \; terms come from when we fill up the group, making STUDENTS choices for the first slot, then STUDENTS-1 choices for the second, and so on. In this way, we end up making _.map(_.range(GROUP), function(l){ return (STUDENTS-l);}).join("\\cdot") = \dfrac{STUDENTS!}{(STUDENTS-GROUP)!} \;. The GROUP! \; term comes from the number of times we've counted a group as different because we chose the students in a different order. There are GROUP! \; ways to order a group of GROUP, so for every group, we've overcounted exactly that many times.

We can use the same logic to count the number of groups that only have ARE_B ? "boys" : "girls".

Specifically, the number of ways to pick a group of GROUP students out of NUM_B is \dfrac{NUM_B!}{(NUM_B-GROUP)!GROUP!} = \binom{NUM_B}{GROUP}.

So, the probability that the teacher picks a group of all ARE_B ? "boys" : "girls" is the number of groups with only ARE_B ? "boys" : "girls" divided by the number of total groups the teacher could pick.

This is \displaystyle \frac{\frac{NUM_B!}{(NUM_B-GROUP)!\cancel{GROUP!}}} {\frac{STUDENTS!}{(STUDENTS-GROUP)!\cancel{GROUP!}}} = \frac{\frac{NUM_B!}{NUM_B-GROUP!}}{\frac{STUDENTS!}{STUDENTS-GROUP!}}

We can re-arrange the terms to make simplification easier \left(\dfrac{NUM_B!}{NUM_B-GROUP!}\right) \left(\dfrac{STUDENTS-GROUP!}{STUDENTS!}\right) = \left(\dfrac{NUM_B!}{STUDENTS!}\right) \left(\dfrac{STUDENTS-GROUP!}{NUM_B-GROUP!}\right)

Simplifying, we get \left(\dfrac{\cancel{NUM_B!}}{_.map(_.range(STUDENTS-NUM_B), function(l){ return (STUDENTS-l); }).join("\\cdot") \cdot \cancel{NUM_B!}}\right) \left(\dfrac{_.map(_.range(STUDENTS-NUM_B), function(l){ return (STUDENTS-GROUP-l); }).join("\\cdot") \cdot \cancel{(NUM_B-GROUP)!}}{\cancel{NUM_B-GROUP!}}\right) = \left(\dfrac{1}{factorial(STUDENTS)/factorial(NUM_B)}\right) \left(factorial(STUDENTS-GROUP)/factorial(NUM_B-GROUP)\right) = \dfrac{PRETTY_NUM}{PRETTY_DEM}

randRange(4,7) randRange(2,COINS-2) random < 0.5 IS_H ? "heads" : "tails" factorial(COINS)/(factorial(NUM)*factorial(COINS-NUM)) Math.pow(2,COINS) getGCD(NUM_RIGHT,NUM_ALL) NUM_RIGHT/GCD NUM_ALL/GCD

If you flip a fair coin COINS times, what is the probability that you will get exactly NUM NAME?

PRETTY_NUMER/PRETTY_DENOM

One way to solve this problem is to figure out how many ways you can get exactly NUM NAME, then divide this by the total number of outcomes you could have gotten. Since every outcome has equal probability, this will be the probability that you will get exactly NUM NAME.

How many outcomes are there where you get exactly NUM NAME? Try thinking of each outcome as a COINS-letter word, where the first letter is "H" if the first coin toss was heads and "T" if it was tails, and so on.

So, the number of outcomes with exactly NUM NAME is the same as the number of these words which have NUM IS_H ? "H's" : "T's" and COINS-NUM IS_H ? "T's" : "H's".

How many of these are there? If we treat all the letters as unique, we'll find that there are COINS! different arrangements, overcounting NUM! times for every time we only switch the IS_H ? "H's" : "T's" around, and COINS-NUM! times for every time we only switch the IS_H ? "T's" : "H's" around. [Show me why]

Let's say we toss a coin 5 times, and get tails three times. How many different re-arrangements are there of the letters "HHTTT"? Well, we have five choices for the first slot, four for the second slot, and so on, resulting in 5\cdot4\cdot3\cdot2\cdot1 = 5! = 120 \; different re-arrangements. But, this treats all the letters as unique, when
HTHTT is the same as
HTHTT, and
HTHTT,
and so on. So really, we need to replace all these different re-arrangements where we only move the tails around with one re-arrangement, HTHTT. There are 3! of these multi- colored arrangements for every normal one, so that means dividing our first guess of 5! by 3!. By the exact same logic, we need to divide by 2! to avoid overcounting every permutation where we just move the heads around. So, the number of re-arrangements is \dfrac{5!}{3!2!} = \binom{5}{3}.

So, there are \dfrac{COINS!}{NUM!COINS-NUM!} = NUM_RIGHT different outcomes where you get exactly NUM NAME. [How many total outcomes are there?]

Well, if you only flip one coin, there are two outcomes, if you flip two there are four outcomes, if you flip three there are eight. Each time you flip another coin, you double the number of possible outcomes.

Altogether, there are 2^{COINS} = NUM_ALL total possible outcomes.

So, the probability that you will get exactly NUM NAME is \dfrac{NUM_RIGHT}{NUM_ALL} = \dfrac{PRETTY_NUMER}{PRETTY_DENOM}.

So, the probability that you will get exactly NUM NAME is \dfrac{NUM_RIGHT}{NUM_ALL}.