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Math
A |
Probability
Involving AND and OR |
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We saw the
connective words AND and OR when we studied logic.
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In logic, we
learned that a sentence "p and q"
is true only when both p and q
are true. |
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In
probability, an outcome is in event "A
and B" only when the outcome is in both event
A and event B. |
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Rule
(for AND): |
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n(A and B)
means the number of outcomes in both A and B.
n(S) means the total number of possible outcomes
A
die is rolled. What is the probability that the number is even and
less than 4?
Event A:
Numbers on a die that are even: 2, 4, 6
Event B: Numbers on a die that are
less than 4: 1, 2, 3
There is only one number (2) that is
in both events A and B.
Total outcomes S: Numbers on a die: 1,
2, 3, 4, 5, 6 (total = 6) |
Answer:
Probability = 1/6 |
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In logic, we
learned that a sentence "p or q"
is true when either (or both) p
or q are true. |
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In
probability, an outcome is in event "A
or B" when the outcome is in either (or both) event
A or event B. |
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Rule
(for OR): |
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The rule for OR takes
into account those values that may get counted more than once when the
probability is determined. Check out the example below.
A
die is rolled. What is the probability that the number is even or
less than 4?
Event A:
Numbers on a die that are even: 2, 4, 6
P(A)=3/6
Event B: Numbers on a die that are
less than 4: 1, 2, 3 P(B)=3/6
P(A and B) = 1/6 (see rule
above)
Answer:
Probability = P(A) + P(B) - P(A and B)
= 3/6 + 3/6 - 1/6 = 5/6
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**Notice
in this problem that the number 2 appears in both event A and
event B. If we did not subtract the P(A and B), the answer
would be 1 - which we know is not true since the number 5 appears
in neither event. |
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