11.2 Probability Using “Not” and “Or”
Example. A single die is rolled. Find the probability that:
(a) the number 3 is rolled
(b) a number other than 3 is rolled
Example. A single die is rolled. Find the probability that:
(c) the number 7 is rolled
(d) a number less than 7 is rolled
Events Involving “Not” (Probability of a Complement)
The probability that event E will not occur is equal to
one minus the probability that it will occur.
𝑃(𝐸 ) = 1 − 𝑃(𝐸 ′ )
Example: In 10.2 and 10.5 we considered a 3-character
PIN formed using the digits 0 – 9.
103 = 1000 possible PINs,
10 × 9 × 8 = 720 without any repeated digits, and
1000 – 720 = 280 with at least one repeated digit.
P(at least one repeated digit) =
Events Involving “Or”
Recall from both chapters 2 and 10 that
𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵) − 𝑛(𝐴 ∩ 𝐵)
𝑛(𝐴 or 𝐵) = 𝑛(𝐴) + 𝑛(𝐵) − 𝑛(𝐴 and 𝐵)
In terms of probability, this results in:
𝑃(𝐴 or 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 and 𝐵)
Example: A single card is drawn from a 52-card deck.
What is the probability that the card is
(a) a heart?
(b) a queen?
(c) a heart and a queen?
(d) a heart or a queen?