Download Example 1 A hat contains 20 tickets, each with a different number

Example 1 A hat contains 20 tickets, each with a different number from 1 to
20. If 4 tickets are drawn at random, what is the probability that the largest
number 1s 15 and the smallest number is 9?
Example 2 On the shelf there is a five-volume book standing at random. What
is the probability that all volumens are standing in the right order?
Example 3 A lift carrying 6 passengers stops at all 9 floors. What is the
probability that
a) all passengers will get out at different floors each?
b) all passengers will get out at the same floor?
Example 4 In tossing a die what is the probability of getting an odd number
or a number less than 4?
Example 5 At a political rally, there are 8 Democrates and 10 Republicans.
Six of the Democrates are females and five of the Republicans are females. If
a person is selected at random, find the probability that the person is a female
or a Democrat.
Example 6 A man is given n keys of which only one fits his door. He tries
them successively (sampling wihout replacement). This procedure may require
1, 2, . . . , n trials. Show that each of these n outcomes has probability n1 .
Example 7 From the population of five symbols a, b, c, d, e a sample of size 25
is taken. Find the probability that the sample will contain five symbols of each
kind.
Example 8 In the box we have 7 black balls and 5 white balls. Choosing
randomly 2 balls find probability that
a) they are of the same colour
b) they are different colour
Example 9 A box contains four $10 bills, six $5 bills, and two $1 bills. Two
bills are taken at random from the box without replacement. What is the probability that both will be of the same denomination?
Example 10 A standard die is rolled with outcomes X1 , X2 , X3 . What is the
probability that X1 ≤ X2 ≤ X3 ?
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Conditional probability
The probability of an event B under the condition that an event A occurs
is called the conditional probability of B given A and is denoted by P (B | A).
We have
P (A ∩ B)
P (B | A) =
.
P (A)
If A, B ⊂ Ω are the events and P (A) 6= 0, P (B) 6= 0 then
P (A ∩ B) = P (A) · P (B | A) = P (B) · P (A | B).
Independent events
The events A, B ⊂ Ω such that P (A ∩ B) = P (A) · P (B) are called independent.
The events A1 , A2 , . . . , Am ⊂ Ω are called independent if
P (A1 ∩ A2 ∩ . . . ∩ Am ) = P (A1 ) · P (A2 ) · . . . · P (Am )
as well for every k different events Aj1 , Aj2 , . . . Ajk where k = 2, . . . , m − 1
P (Aj1 ∩ Aj2 ∩ . . . ∩ Ajk ) = P (Aj1 ) · P (Aj2 ) · . . . · P (Ajk ).
Law of total probability, Baye’s theorem
Assume that the sample space Ω is partitioned into n subsets
A1 , A2 , . . . , An satisfying:
a) Ai ∩ Aj = ∅ for any i, j = 1, . . . , n, i 6= j
b) Ai 6= ∅ for i = 1, . . . , n
c) A1 ∪ A2 ∪ . . . ∪ An = Ω
then for any event E ⊂ Ω we have
P (E) = P (A1 ) · P (E | A1 ) + P (A2 ) · P (E | A2 ) + . . . + P (An ) · P (E | An )
P (Ai | E) =
P (Ai ) · P (E | Ai )
P (E)
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Bernoulli trials
Repeated independent trials are called Bernoulli trials if there are only
two possible outcomes for each trial and their probabilities remain the same
throughout the trials.
Theorem
Let b(k; n, p) be the probability that n Bernoulli trials with probabilities p
for success and q = 1 − p for failure result in k successes (0 ≤ k ≤ n). Then
!
n k
b(k; n, p) =
p (1 − p)n−k .
k
Example 11 In tossing a coin 3 times what is the probability of getting 2
heads if we know that was tail on the last coin?
Example 12 Three dice are rolled. If no two show the same face, what is the
probability that on one is a ”six”.
Example 13 From a deck of five cards numbered 2, 4, 6, 8 and 10, respectively,
a card is drawn at random and replaced. This is done three times. What is the
probability that the card numbered 2 was drawn exactly two times, given that
the sum of the numbers on the three draws is 12?
Example 14 Suppose that 5 man out of 100 and 25 women out of 10000 are
colorblind. A colorblind person is chosen at random. What is probability of his
being male? (Assume males and femals to be in equal numbers).
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