RULE: For any event 𝑬, its probability is
between 0 and 1, inclusive.
In precise math language:
0 ≤ 𝑃(𝐸) ≤ 1
Why? Since 𝑃(𝐸) =
ℎ𝑜𝑤 𝑚𝑎𝑛𝑦 𝑡𝑖𝑚𝑒𝑠 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
,
ℎ𝑜𝑤 𝑚𝑎𝑛𝑦 𝑖𝑡𝑒𝑚𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒,𝑡𝑜𝑡𝑎𝑙
The numerator is always going to be at least 0 but less than the denominator, so the value of
𝑃(𝐸) cannot be negative and it cannot be greater than 1.
Example: Rolling one die
The sample space { 1, 2, 3, 4, 5, 6 }
An example of some event 𝐸 =Rolling a small number, a 1 or a 2
𝑃(𝐸) =
RULE: If 𝐸 is an impossible event, an event
that cannot ever occur, then 𝑃(𝐸) = 0,
the probability of 𝐸 is zero.
RULE: If 𝐸 is a certain event, guaranteed to
occur, then 𝑃(𝐸) = 1,
the probability of 𝐸 is 1.
Complementary Event – if 𝐸 is some event,
then the complementary event is defined as
“not event 𝐸” and we write it as the symbol 𝐸̅
2 1
=
6 3
𝐸 = rolling a 7.
0
𝑃(𝐸) = = 0
6
𝐸 = rolling a number between 0 and 10
6
𝑃(𝐸) = = 1
6
If 𝐸 = rolling a 5
1
𝑃(𝐸) =
6
Then 𝐸̅ = rolling something other than 5.
𝑃(𝐸̅ ) =
RULE: 𝑃(𝐸̅ ) = 1 − 𝑃(𝐸)
The probability of the complementary event is
equal to 1 minus the probability of the event.
RULE: The probabilities of all the outcomes in
the sample space total exactly precisely 1.
∑
𝑃(𝐸) = 1
Example: Drawing one card
{ the fifty-two cards listed out }
𝐸 =drawing a face card
𝑃(𝐸̅ ) = 1 −
𝑃(𝐸) =
12
3
=
52 13
𝐸 =drawing a green card
0
𝑃(𝐸) =
=0
52
𝐸 =drawing a red card or a black card
52
𝑃(𝐸) =
=1
52
If 𝐸 = drawing a diamond,
13 1
𝑃(𝐸) =
=
52 4
Then 𝐸̅ = drawing a non-diamond (a heart,
club, or spade)
5
6
𝑃(𝐸̅ ) =
1 5
=
6 6
𝑃(1) + 𝑃(2) + 𝑃(3) + 𝑃(4) + 𝑃(5) + 𝑃(6)
1 1 1 1 1 1
= + + + + + =1
6 6 6 6 6 6
𝑒𝑎𝑐ℎ 𝐸
Adapted from [JK10 Section 4.2] and [Blu4 Section 4-2]
39 3
=
52 4
𝑃(𝐸̅ ) = 1 −
∑
1 3
=
4 4
𝑃(𝑐𝑎𝑟𝑑)
𝑒𝑎𝑐ℎ 𝑐𝑎𝑟𝑑
=
1
1
1
52
+
+ ⋯+
=
=1
52 52
52 52
Документ1
4/11/2020 2:15 AM
- D.R.S