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Combined Events
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Venn diagrams, intersection and union of events, conditional probability, independent events
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Sample space diagrams
SL 4.6
A sample space diagram is a table representation for scenarios where two probability events occur side by side, and we're interested in certain combinations of those events.
For example, consider a game where a player rolls a die and then flips a coin. If the coin lands on heads, we add 1 to their die roll, and if it lands on tails we double their die roll.
Then we can answer questions like:
What is the probability of scoring higher than 6?
To do this, we simply count the number of cells that are greater than 6 and divide by the total number of cells. So in this case, we have the cells with 7,8,10,12, so the answer is
124=31
Venn Diagrams
4. Prior learning
A Venn diagram is a visual tool used to illustrate relationships between sets or events. Each event is represented by a circle, and overlaps between circles represent shared outcomes. All circles lie within the larger universe U, which is the whole sample space.
Each region in the Venn diagram can be numbered to indicate the number of elements that fall into that category. For example:
the 7 in the diagram above represents elements that are in C, but at the same time not in A or B.
the 4 represents elements that are in A and B,m but not in C.
Intersection of probabilities
SL 4.6
The intersection is the event where both A and B occur simultaneously, denoted A∩B.
On a Venn diagram, this is the shared region between the two sets:
Union of probabilities
SL 4.6
The union is the event that at least one of A or B occurs.
The union is denoted A∪B and has probability
P(A∪B)=P(A)+P(B)−P(A∩B)📖
The reason for the subtraction −P(A∩B) is that adding P(A)+P(B) counts the shared region of A and B twice.
Calculating probabilities from Venn diagrams
SL 4.6
Venn diagrams are often filled in with numbers representing the number of samples in each category. Probabilities of certain events can be found from Venn diagrams by counting the number of relevant outcomes, and dividing by the sum of all the values in the Venn diagram.
For example, the probability of A or B, but not both, is represented by regions on the Venn diagram alongside.
The number of elements in those regions is
5+1+6+2=14
and the total number of elements is
8+5+4+2+1+3+6+7=36.
So the probability of a randomly selected element being in one of those regions is
3614=187
Mutually exclusive events
SL 4.6
Events are mutually exclusive if they cannot both occur at once. In this case, the intersection probability is zero:
P(A∩B)=0🚫
And therefore
P(A∪B)=P(A)+P(B)📖
Conditional Probability
SL 4.6
Conditional probability is the probability of event A happening given we already know event B has occurred. It's calculated by taking the probability that both events occur, divided by the probability of the known event B:
P(A∣B)=P(B)P(A∩B)📖
Notice that we can rearrange this formula to get a general formula for the probability of multiple events,
P(A∩B)=P(A∣B)P(B)=P(B∣A)P(A)
Independent events
SL 4.6
If two events A and B are independent, then knowing whether or not one happened gives no information on whether or not the other happened, and
P(A∣B)=P(A),P(B∣A)=P(B)
Rearranging the conditional probability formula gives the fact that for independent A and B,
P(A∩B)=P(A)×P(B)
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