Revision for Midterm 2
Chapter 4: Probability
(1) Methods of Assigning Probabilities:
(a) Classical Method: when the probabilities are assigned based on laws or rules.
In this case, the probability of an event Ein a sample space Sis
P(E) = #(E)
#(S)·
Examples:
•Probability to draw a card of hearts from a 52-card deck.
•Probability that at least two students in the class celebrate their anniversary
on the same day.
(b) Relative Frequency of Occurrence: when the assignment of probabilities is based on
historical data.
In this case, the probability of an event Ein a sample space Sis
P(E) = Number of times the event Ehas occurred
Total number of opportunities for the event to occur·
Examples:
•Probability to be hit by a lightning.
•Probability to get injured at work.
(c) Subjective probability: when the assignment of probabilities is based on feelings,
knowledge and/or past experiences of a person or a group of people.
Examples:
•A financial advisor tells there are 75% chance that a major financial crisis
happens within the next five years.
•A taxi driver asserts there are 80% chance a route will be faster than another
one.
(2) Counting the Probabilities:
Different methods for computing the numbers of elements in Eand S:P(E) = #(E)
#(S).
(a) The mn-Counting Rule: Suppose a procedure can be realized in a sequence of k
(independent) steps. If the i-th step offer nipossibilities, for all i= 1,2, . . . , n, then
the procedure can be realized in
n1n2n3· · · nk
different ways.
Example:
In a certain country, all license plates consist of three letters (A to Z), followed by
three digits (0 to 9). How many different plates are there?
Solution: There are 26 ∗26 ∗26 ∗10 ∗10 ∗10 = 17,576,000 different plates.
(b) Combinations: Suppose there are ndifferent items. The number of ways to select
r(with 0≤r≤n) different items is
nCr=n!
r!(n−r)!·
Examples:
•A box contains 30 chips, 5 of them being defective. If 10 of these 30 chips are
selected, in how many ways can exactly 3 of the 5 defective chips be selected?
Solution: This can be seen as a 2-step process; we select 3 of the 5 defective
chips (5C3), and then 7 of the 25 intact ones (25C7). By the mn-Counting, this
can be done in 5C3·25 C7= 4,807,000 ways.
•Find the number of distinguishable permutations of the given letters MISSIS-
SIPPI”.
Solution: Writing an anagram with the letters of the word MISSISSIPPI con-
sists in filling 11 positions in 4 steps:
- Choose 1 position for the M: 11 C1= 11 possibilities
- Choose 4 positons among the 10 remaining positions for the I’s: 10C4= 210
possibilities
- Choose 4 positions among the 6 remaining positions for the S’s: 6C4= 15
possibilities
- Choose 2 positions among the 2 remaining positions for the P’s: 2C2= 1
possibilities
By the mn-Counting Rule, this can be done in 11∗210 ∗15∗1 = 34,650 different
ways.
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