Independent and Mutually Exclusive Events Quiz
The question sheet
Reveal any answer as you study-
Which equation shows that two events A and B are independent?
- P(A) + P(B) = 1
- P(A ∩ B) = 0
- P(A ∩ B) = P(A)P(B)
- P(A ∩ B) = P(A) + P(B)
Reveal answer
Answer: P(A ∩ B) = P(A)P(B)
Source evidence
PDF page 146: • P(A ∩ B) = P(A)P(B)
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Two events A and B are mutually exclusive if:
- P(A ∩ B) = P(A)P(B)
- They cannot occur at the same time
- One occurring affects the other
- Sampling is with replacement
Reveal answer
Answer: They cannot occur at the same time
Source evidence
PDF page 148: A and B are mutually exclusive events if they cannot occur at the same time. Said another way, If A occurred then B cannot occur and vise-a-versa. This means that A and B do not share any outcomes and P(A ∩ B) = 0 .
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For mutually exclusive events A and B, what is P(A ∩ B)?
- 0
- 0.5
- P(A)P(B)
- 1
Reveal answer
Answer: 0
Source evidence
PDF page 148: A and B are mutually exclusive events if they cannot occur at the same time. Said another way, If A occurred then B cannot occur and vise-a-versa. This means that A and B do not share any outcomes and P(A ∩ B) = 0 .
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When sampling is done with replacement, the events are considered:
- Conditional
- Independent
- Dependent
- Mutually exclusive
Reveal answer
Answer: Independent
Source evidence
PDF page 146: of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.
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With replacement, each member of a population:
- Changes the sample space
- Cannot be chosen
- Can be chosen only once
- May be chosen more than once
Reveal answer
Answer: May be chosen more than once
Source evidence
PDF page 146: • With replacement: If each member of a population is replaced after it is picked, then that member has the possibility
PDF page 146: of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.
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Without replacement, each member of a population may be chosen:
- Never
- Only once
- More than once
- Twice
Reveal answer
Answer: Only once
Source evidence
PDF page 146: • Without replacement: When sampling is done without replacement, each member of a population may be chosen
PDF page 146: only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent. If it is not known whether A and B are independent or dependent, assume they are dependent until you can show otherwise.
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Do 'independent' and 'mutually exclusive' mean the same thing?
- Yes, always
- No, they differ
- Only for coins
- Only with replacement
Reveal answer
Answer: No, they differ
Source evidence
PDF page 146: Independent and mutually exclusive do not mean the same thing.
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For P(A)=0.4, P(B)=0.2, P(A∩B)=0.08, are A and B independent?
- Only if disjoint
- No
- Yes
- Cannot be determined
Reveal answer
Answer: Yes
Source evidence
PDF page 150: Suppose P(A) = 0.4 and P(B) = 0.2 . P(A ∩ B) = 0.08 . Are events A and B independent? Hint: You must show
PDF page 150: • P(A ∩ B) = P(A)P(B)
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Given P(G)=0.6, P(H)=0.5, P(G∩H)=0.3, are G and H independent?
- No
- Mutually exclusive
- Yes
- Only sometimes
Reveal answer
Answer: Yes
Source evidence
PDF page 151: P⎝G⎠P⎝H⎠ = ⎝0.6⎠(0.5⎠ = 0.3 = P⎝G ∩ H⎠
PDF page 150: • P(G ∩ H) = P(G)P(H)
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For C and D with P(C∩D)=0.225 not zero, C and D are:
- Equal
- Mutually exclusive
- Not mutually exclusive
- Complementary
Reveal answer
Answer: Not mutually exclusive
Source evidence
PDF page 151: b. No, because P(C ∩ D) is not equal to zero.
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Why are being female and having long hair not independent?
- P(F∩L) = 0
- P(F) = P(L)
- They are mutually exclusive
- P(F∩L) does not equal P(F)P(L)
Reveal answer
Answer: P(F∩L) does not equal P(F)P(L)
Source evidence
PDF page 153: Solution 1 P(F ∩ L) = P(F)P(L) . We are given that P(F ∩ L) = 0.45 , but Check whether P(F)P(L) = (0.60)(0.50) = 0.30 . The events of being female and having long hair are not independent because P(F ∩ L) does not equal P(F)P(L) . Solution 2 Check whether P(L|F) equals P(L) . We are given that P(L|F) = 0.75 , but P(L) = 0.50 ; they are not equal. The events of being female and having long hair are not independent. Interpretation of Results The events of being female and having long hair are not independent; knowing that a student is female changes the probability that a student has long hair.
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For die faces, C={3,5} and E={1,2,3,4}. Are C and E mutually exclusive?
- No
- Only if fair
- Yes
- Cannot tell
Reveal answer
Answer: No
Source evidence
PDF page 150: Are C and E mutually exclusive events? (Answer yes or no.) Why or why not?
PDF page 150: No. C = {3, 5} and E = {1, 2, 3, 4}. P C ∩ E = . To be mutually exclusive, P(C ∩ E) must be zero.
Introductory Business Statistics
Introductory Business Statistics by OpenStax, used under CC BY 4.0. Changes made by Stratacademy.
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