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Tree Diagrams
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Tree diagrams, selection with/without replacement, dependence
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Selection with & without replacement
SL 4.6
In probability, a selection is the action of choosing one or more items from a set or group. We can perform selections either with replacement or without replacement.
In selection with replacement, each chosen item is returned to the original group before the next choice, keeping the probabilities constant across selections.
In selection without replacement, the chosen items are removed from the group, causing probabilities to change after each pick because the number of available items decreases.
This difference significantly impacts how probabilities are calculated, especially in problems involving multiple selections.
Tree Diagram
SL 4.6
A tree diagram is a map of what can happen, one step at a time. We use them in probability scenarios with multiple steps, mostly when the result of one step affects the probability of the next step.
To explain, consider this example: We have a bag with 3 red marbles and 2 green marbles. You draw one marble from the bag, put it in your pocket, and then draw a second marble from the bag.
The diagram starts at the point where nothing has happened yet, and the branches show the different possibilities for what can happen next.
There are two branches leaving from the start: one goes to "G" (green), the other to "R". They are numbered with the probability of the first marble being a certain color. Since there are 3 red marbles and a total of 5 marbles in the bag, that branch has probability 53.
Now the tree has split into two possibilities, and for each of those there are two possibilities for what can happen next. But those probabilities depend on what the first marble was, because if we drew a green marble there is only 1 green marble left.
The probabilities on the next four branches are based on where you already are in the tree. For example, the 41 in the top right is the probability that the second marble is green if the first marble was green.
The probability of drawing two green marbles can be found by multiplying the branches that lead from "start" to two greens:
P(GG)=52×41=101
The probability of drawing two different color marbles (in any order) requires adding the probability of RG and the probability of GR:
P(different)=52×43+53×42=206+206=2012=53
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