Explanation (below)
This lesson centered on the experiment of drawing two chips out of a bag that contained both green and yellow chips. We did two kinds of experiments this way:
The first kind is drawing chips with replacement. That means that after you draw the first chip, you put it back in the bag before drawing the second chip.
If you do this with two green and two yellow chips, you get that the probability of drawing a green is 2/4 the first time, and 2/4 the second time--the probability doesn't change.
Likewise, the probability of drawing a yellow is 2/4 the first time and 2/4 the second time. You can use those probabilities to find the probabilities of a 2-chip combination in 3 ways:
Using a tree diagram
In a tree diagram the first branches (blue) show the first chip you pick
The second set of branches (red) show the second chip you pick
Find the probability of any particular combination by multiplying along the
branch
Find the probability of one of each by adding the two ways that can happen
The probabilities in simplified form are:
P(GG)=1/4
P(YY)=1/4
P(1 of each)=1/2
Using multiplication
This works just like the tree diagram method for finding P(GG) and P(YY). Finding P(1 of each) can be done by the tree diagram method, or by subtracting P(GG) and P(YY) from 1
The probabilities in simplified form are:
P(GG)=1/4
P(YY)=1/4
P(1 of each)=1/2
Using a square area diagram
In this square the vertical lines show the the probabilities of the first chip
you pick
The horizontal lines show the second chip you pick
You find the probabilities by counting equal sized squares/rectangles to make
a fraction
The probabilities in simplified form are:
P(GG)=1/4
P(YY)=1/4
P(1 of each)=1/2
The second kind is drawing chips without replacement. That means that after you draw the first chip, you keep if out of the bag while drawing the second chip.
If you do this with two green and two yellow chips, you get that the probability of drawing a green is 2/4 the first time, but then there are only 3 chips left, so the probability of getting another green is 1/3 and the probability of getting a yellow is now 2/3.
Likewise the probability of drawing a yellow is 2/4 the first time, but then there are only 3 chips left, so the probability of getting another yellow is 1/3 and the probability of getting a green is now 2/3
Using a tree diagram
In a tree diagram the first branches (blue) show the first chip you pick
The second set of branches (red) show the second chip you pick
Find the probability of any particular combination by multiplying along the
branch
Find the probability of one of each by adding the two ways that can happen
The probabilities in simplified form are:
P(GG)=1/6
P(YY)=1/6
P(1 of each)=2/3
Using multiplication
This works just like the tree diagram method for finding P(GG) and P(YY). Finding P(1 of each) can be done by the tree diagram method, or by subtracting P(GG) and P(YY) from 1
The probabilities in simplified form are:
P(GG)=1/6
P(YY)=1/6
P(1 of each)=2/3
Using a square diagram
In this square the vertical lines show the the probabilities of the first chip
you pick
The horizontal lines show the second chip you pick
You find the probabilities by counting equal sized squares/rectangles to make
a fraction
The probabilities in simplified form are:
P(GG)=1/6
P(YY)=1/6
P(1 of each)=2/3