Intersection of Events
•The intersection of events A and B, denoted A ∩ B, is the collection of all outcomes that are elements of both of the sets A and B.
•It corresponds to combining descriptions of the two events using the word “and.”
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Examples: Q) In the experiment of rolling a single die. • Find the intersection E ∩ T of the events E: “the number rolled is even” and T: “the number rolled is greater than two.” -->Solution: • The sample space is S={1,2,3,4,5,6}. Since the outcomes that are common to E={2,4,6} and T={3,4,5,6} are 4 and 6, E∩T={4,6}. • “the number rolled is even and is greater than two.” • The only numbers between one and six that are both even and greater than two are four and six, corresponding to E ∩ T given above. |
Q) • A single die is rolled.
1. Suppose the die is fair. Find the probability that the number rolled is both even and greater than two.
2. Suppose the die has been “loaded/altered” so that P(1)=1⁄12, P(6)=3⁄12, and the remaining four outcomes are equally likely with one another. Now find the probability that the number rolled is both even and greater than two
• Solution:
• In both cases the sample space is S={1,2,3,4,5,6} and the event in question is the intersection E∩T={4,6} of the previous example. P(E∩T)=2⁄6.
• The information on the probabilities of the six outcomes that we have so far is • Since P(1)+P(6)= 4⁄12 = 1⁄3 and the probabilities of all six outcomes add up to 1,
• P(2)+P(3)+P(4)+P(5)=1−1/3=2/3
• Thus 4p=2⁄3, so p=1⁄6. In particular P(4)=1⁄6.
• Therefore P(E∩T)=P(4)+P(6)=1/6+3/12=5/12
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Union of two events
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Q) In the experiment of rolling a single die, find the union of the events E: “the number rolled is even” and T: “the number rolled is greater than two.”