Concept explainers
(a)
To find: the probability that all three people have different birthday.
(a)
Answer to Problem 45E
The probability is
Explanation of Solution
Given information:
There are three persons in a room and there are total
Suppose the birthday of one person is known.
If second person has a different birthday, then there are
Thus, the probability that person 2 has a different birthday from person 1 is
If third person has a different birthday from the person 1 and 2, then there are
Thus, the probability that person 3 has a different birthday from person 1 and 2 is
So, the probability that all 3 have different birthdays is
Therefore, the probability is
(b)
To find: the probability that two people have same birthday.
(b)
Answer to Problem 45E
The probability is
Explanation of Solution
Given information:
There are three persons in a room.
As it is known from part (a) that the probability of all they have different birthday is
So, the probability of any two people has same birthday is
Therefore, the probability is
(c)
To find: the probability that four people have shared birthday.
(c)
Answer to Problem 45E
The probability is
Explanation of Solution
Given information:
There are three persons in a room.
Let the probability can be represented by
If fourth person added in the room. Then, that person has a different birthday from the other three such that
The probability that none of four people have the same birthday is
And the probability that at least two of four people have the same birthday is
Substitute
Therefore, the probability is
(d)
To find:whether the result is reasonable to assume that all calendar dates are equally likely birthdays or not.
(d)
Answer to Problem 45E
The result is not reasonable to assume that all birthdays are equally likely.
Explanation of Solution
Given information:
There are three persons in a room.
There are two factors that need to be taken into account. First, if checking the actual data, there will be an uneven distribution of birthdays. For instance, in the United states, births during the months of July, August, and September are more common than in other months. Second, February
Therefore, the result is not reasonable to assume that all birthdays are equally likely.
Chapter 3 Solutions
Calculus: Graphical, Numerical, Algebraic: Solutions Manual
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