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Combined Events
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Venn diagrams, intersection and union of events, conditional probability, independent events
Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)
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
SL 4.6
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.
Venn diagrams are often filled in with numbers representing the number of samples in each category.
Intersection of probabilities
SL 4.6
The intersection is the event where both A and B occur simultaneously, denoted A∩B.
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)📖
This formula is sometimes referred to as the inclusion-exclusion rule. It is often rearranged in the form
P(A∩B)=P(A)+P(B)−P(A∪B)🚫
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)
Nice work completing Combined Events, here's a quick recap of what we covered:
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