Jefferson's Scribe - Introduction to Probability
大家好(hello everyone), my name is Jefferson, thanks to Janet I am the scribe for today. Today Mr K started us off with some questions on the board. The first one went something like this:
How many ways can you get from A to B by the shortest route possible? (i.e. only moving right and down)
There are many routes to take that are short. And out of all the shortest routes you can think of to take, they all end up 4 blocks to the left and 3 blocks down.
And resulting in "RRRRDDD". With this we were able to figure out a solution for this question. Manny came up to the board and explained to us that this question is similar to out "MISSISSIPPI" problem. Since there are 11 letters in mississippi,4 "S"s, 4 "I"s and 2 "p"s, we plug those numbers into the "permutations of non-distinguishable objects" formula n!/(k1! k2! k3!). And it will look something like this:
11!/(4! 4! 2!).
SOOOOOOO back to our original question. There are 7 letters in (RRRRDDD), 4 "R"s and 3"D"s, lets plug them into the formula and see what happens!!!
7!/( 4! 3! ) = 7 * 6 * 5 * 4 * 3! / (4 * 3 * 2 * 1)3!
7!/( 4! 3! ) = 7*5
7!/( 4! 3! ) = 35
*"3!" get cancelled out
*4 gets cancelled out
* 6 / 3 = 2
* 2 / 2 = 1
So there are 35 ways to get from point A to point B.
And the next question had a similar diagram:
To find the solution we had to find the number of was to get from
point A to point a, point a to point b, and point a to point B.
A-a : RRDD 4!/(2!2!) = 6
a-b : Rd 2! = 2
b-B : RRRRDDD 7!/(4!3!) = 35
6 * 2 * 35 = 420 ways
Mr K then showed us another way which he liked to call "The inside secrets of the universe".
Do we all know why there are a bunch of 1s at the top and left? Because from point A there is only one way to get there.
By adding the top right corner and the bottom left corner of a single square, you'll get an answer for the bottom right corner of the square. And that answer determines how many ways are possible coming from point A to that bottom right corner. In this case : 1 + 1 = 2. Our bottom right corner is 2 and that means there are 2 ways that point(2) from point A.
And by adding all the corners until you get to the last corner of the entire diagram. The answer is 35!! Just like our first solution!
And for the next question:
The answer is also 420!!
The method that Mr K showed us was called "Pascal's triangle" Which is used everywhere!!
And the next and final question:
Write your answer to this question as a fraction only. Do not discuss with neighbor.
What is the probability that the sky will fall?
The solutions that most of the class answered were the following.
1/(10)^10^100 ' 1 out of googolplex things that could happen
o / ∞
0/0 (4 students answered) 'means that it is impossible to happen
1/2 (3 students answered) '50% chance of it happening
1/ ∞ (3 students answered) '1 out of infinite things that could happen
1/ 10^10^12 '1 out of a trillion things that could happen
The formula for this sort of question would be:
# of favorable outcomes/everything that could happen
And the solution is that the sky is undefined
The reason why Mr K gave us this question was to illustrate the concept of a sample space
If you were to flip a coin, the probability of getting heads or tails is 2. If you were to flip the coin again the probability 2 again. This is explained by this tree diagram
This is an easy way to find the probability of a coin
If you were to roll a six sided die the chance of getting a 6 is 1 out 6 chances and 0 out of 6 chances of getting a seven.
If you were to roll 2 dice you would get 1 out of 36 ways of getting a number.
This chart illustrates the probability of getting any number from rolling 2 dice
i.e there are 6 chances of rolling a five.
Homework is on p 371 of the text book # 1-13 (all)
WOOOOO I'M DOWN!!! This took me about 4 hrs to finish this scribe...man talk about dedication...hah I'm kidding! Now, the next scribe is non other than.... LATER DAYS,