Basic Rules of Probability
Section 3.3
Probability Rules:
1. P(
) = 0 *The
smallest possible probability is of an impossible event (null set
).
2. P(S) = 1 *The largest possible probability is of a certain event (event equal to S, the sample space
3. 0
P(E)
1
** Probabilities exist between 0 and 1, inclusive.
**If you ever get a negative probability or a probability greater than 1, recheck you work and find your mistake.
Mutually exclusive events: Two events that cannot both occur at the same time.
i.e. E
F =
if and only if E and
F are
mutually exclusive.
Example 1:
A Pair of dice is rolled.
Notice that sample space is not = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
The sample space is all possible outcomes of two dice rolled.
Sample space has 36 elements. (see page 137)
Find the probability that:
a) the sum is less than 5
b) the sum is odd
c) the sum is less than 9 and odd.
d) The sum is less than 9 or odd
a) P(a) = n(a)/n(S) = 6/36 = 1/6
b) P(b) = 18/36 = ½
c) P(c) = 12/36 = 1/3
d) Use complement: the sum is greater than 9 and even = 4/36 = 1/9. So
1 – 1/9 = 8/9
Notice all of the probabilities occur between 0 and 1.
More Probability Rules.
4.
P(E
F) = P(E) + P(F) – P(E
F)
5. If E and F are mutually exclusive:
P(E
F) = P(E) + P(F)
6. P(E) + P(E’) = 1
or
1 – P(E) = P(E’)
or
1 – P(E’) = P(E)
Example 2: Find the odds from EX 1.
a) n(a) : n(a’) = 6 : 30 = 1 : 5
b) n(b) : n(b’) = 18 : 18 = 1 : 1
c) n(c) : n(c’) = 12 : 24 = 1 : 2
d) n(d) : n(d’) = 32 : 4 = 8 : 1
#59 requires a lot of thought. Use the Rules of Probability!
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