Find P(Ac).
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Find P(Ac).
Answer: P(Ac) = 1 - P(A) = 0.4
Find P(A and Bc).
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Find P(A and Bc).
Answer: P(A and Bc) = P(A) - P(A 
B) = 0.6 - 0.2 = 0.4
Find P(B and Ac).
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Find P(B and Ac).
Answer: P(B and Ac) = P(B) - P(A B) = 0.5 - 0.2 = 0.3
Find P(A or B).
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Find P(A ∪ B).
Answer: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.6 + 0.5 - 0.2 = 0.9
Independent Versus Mutually Exclusive
Remark: Independent is very different from mutually exclusive.
In fact, mutually exclusive events are dependent. If A and B are mutually exclusive events, there is nothing in A and B, and thus:
P(A and B) = 0 ≠ P(A) P(B)
From an urn with 6 red balls and 4 blue balls, two balls are picked randomly without replacement. Find the probability that both balls picked are red.
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From an urn with 6 red balls and 4 blue balls, two balls are picked randomly without replacement. Find the probability that both balls picked are red.
From an urn with 6 red balls and 4 blue balls, two balls are picked randomly without replacement. Find the probability that both balls picked are red.
Answer:
P (both balls picked are red)
=P({first ball red} ∪ {second ball red})
=P(first ball red) P(second ball red | first ball red)
Let A and B be the following two happy events.
A: get a job, B: buy a new car.
It is a given that P(A) = 0.9, P(B) = 0.7. What is the probability of double happiness: that you get a job and buy a new car? In other words, we want to find P(A and B).
Work out your answer first, then click the graphic to compare answers.
Let A and B be the following two happy events.
A: get a job, B: buy a new car.
It is a given that P(A) = 0.9, P(B) = 0.7. What is the probability of double happiness: that you get a job and buy a new car? In other words, we want to find P(A ∩ B). Work out your answer first, then click the graphic to compare answers.
Answer: There is not yet enough information to answer the question.
First, we will ask whether A, B are independent. In this case, the simplistic approach of saying that the two events are independent is not realistic. Thus, we will think harder and try to assess either P(A | B) or P(B | A)? Thinking about it, it is not hard to assess the probability of buying a new car knowing that he/she gets a job. For example, if one thinks that P(B | A) = 0.75 (this probability is subjectively chosen and may be different for different individuals), the person happens to think that the chance to buy a new car knowing that he/she gets a job is 75%.
P(A ∪ B) = P(A) P(B|A) = (0.9)(0.75) = 0.675