3.1 Necesidades formativas
3.1.1 Cursos y capacitaciones
Let A1 be the event that the first toss lands heads, A2 the event that the second toss lands, and A3 the event that the third toss lands heads.
A1 A2 A3 is the event that at least one head occurs.
We are required to compute P(A1 A2 A3) The complement of A1 A2 A3 is A1c n A2c n A3c. That is, the first does not land heads, the second does not and the third does not.
A1 c n A2
c n A3
c = {T, T, T}.
Thus
P(A1c n A2c n A3c) = 1/8.
Hence
P(A1 A2 A3) = 1 – 1/8 = 7/8.
2.9 If A1 A2, then P(A1) ≤ P(A2)
A2 = A1 (A2/A1).
Since A1 and A2/A1 are mutually exclusive, we have P(A2) = P(A1) + P(A2/A1)≥ P(A1) Since P(A2/A1) ≥ 0.
2.9 If A1, A2,…, An are n events, then
= From de Morgan’s law, we have
= hence
=
the probability that an event A will occur “conditional on” the knowledge that another event B has occurred.
Suppose a fair die is rolled and it is known that an even number appeared uppermost. Let A be the event that the number was greater than 3 and B the event that the number that appeared was even. The problem is to find the conditional probability that the event A occurred given that the event B has occurred, P(A|B). since we know that the number was even, the number must be either 2,4 or 6. Therefore, the conditional sample space contains 3 elements. The event A occurs if the number showing is 4 or 6, thus
P(A|B) = 2/3.
We can therefore define conditional probability of A given Bas the number of ways AnB can occur divided by number of elements in the conditional sample space. That is,
,
Where ≠ 0, where ΩB is the condtional sample space given that B has occurred.
Divide the numerator and denominator by nΩ =
In the above example,
n(A B) = 2 n(B) = 3 Definition 2.4
Let A and B be two events such that P(B) > 0. Then the conditional probability of A given B, denoted by P(A|B) is defined to be P(A|B) = 2.2
Examples
2.11 Suppose a box contains 4 red balls and 3 black balls. Compute the probability that (i) the second ball drawn is red if the first ball drawn was red; without
replacement,
(ii) the second ball drawn is red if the first ball drawn was black
Solution
If the first ball drawn is red, there remains 6 balls, 3 red balls and 3 black balls.
The probability of the second ball being red is 63=12. But if the first ball is black, the box is left with 4 red and 2 black so the probability of the second ball being red is then 4/6 = 2/3. Thus
P(2nd is re/first was red) = 3/6/ = ½ P(2nd is red/ first was black) = 4/6 = 2/3.
This shows that the probability of the event “the second ball drawn is red” depends on the colour of the first ball drawn.
2.12 Suppose two fair dice are rolled. If the sum of the numbers appearing is 6, what is the probability that one of the number is 2?
Solution
Let A be the event “one of the numbers is and B the sum is 6. there are five ways for the event B to occur: (3,3), (2,4), (4,2), (5,1) and (1,5) and there are two ways for the event AnB to occur: (2,4) and (4,2).
Thus,
36 5 ) ( , 36 2 )
(A∩B = P B =
P
Hence,
. 5 ) 2
( ) ) (
( Ι = ∩ =
B P
B A B P
A P
2.13 There are two children in a family. If there is at least a girl in this family, what is the conditional probability that both are girls.
Solution The sample space is
} , , ,
{BB GB BG GG
= Ω
Let A be the event “both children are girls and B ”a least a girl in the family.
B = {GB, BG, GG}, A = {GG}, A∩B = {GG}.
P(AΙB) = .
3 1 4 3
4 1 ) (
)
( = =
B P
AnB P
2.14 There are three children in a family. If there is at least one boy and at most two boys in this family. What is the conditional probability that there are exactly two boys in this family.
The sample space is
Ω = {BBB, BBG, BGG, BGB, GBB, GBG, GGB, GGG}
Let B be the event “at least one boy and at most 2 boys in the family” and let A be the event “exactly two boys in the family”. Then
B = {BBG, BGG, BGB, GBB, GBG. GGB}
A∩B = {BBG, BGB, GBB}.
Therefore
P(AΙB) = .
2 1 6 3 8 6
8 3 ) (
)
( = = =
B P
AnB P
Exercises 2.1
1. A fair die is thrown twice.
(i) If it is known that the sum of the numbers appearing was 8, what is the probability that the difference between the two numbers was 2.
(ii) If it is known that the difference the two numbers was 3, what is the probability that the sum of the two numbers was 7?
2. Two unbiased dice are thrown once. What is the probability that (i) at least one 5 is thrown.
(ii) the sum is 10
(iii) the sum is 10 given that no 5 is thrown?
3. Suppose events A and B are sun that P(A) = 1/5, P(AnB) = 1/6.
Find (i) P(BIA); (ii) P(Ac∪Bc).
4. If A and B are two events defined on the same probability space, show that: (i) P(A) = P(A∩B) + P(A ∩ Bc) = P(B) + P(A ∩ Bc) – P(Ac ∩ B).
5. Prove that P(Ac ∩ Bc) = 1 – P(A) – P(B) + P(A ∩ B).
6. Suppose a well balanced coin is tossed twice.
Find the conditional probability that
(i) both coins show a tail given that the first shows a head;
(ii) both are heads given that at least one of them is a head.
7. A red die and a green die are rolled once. Find the conditional probability that:
(i) the number on red die is odd, given that the sum of the two numbers showing is 9;
(ii) the sum of the two numbers is 9 given that one of the numbers is odd and the other even?