Respuesta :
Answer:
The answers to the question are
a) [tex]\frac{1}{7}[/tex]
b) Where
n >7 the probability = 1
n = 7 it is 0.99388
n = 6 it is 0.95716
n = 5 it is 0.850062
n = 4 it is 0.65
n = 3 it is 0.38776
n = 2 it is 0.14286
n = 1 it is 0
c) n > 3 where n is an integer
Step-by-step explanation:
The probability for the first person to be born on a certain day is 1
P(Week Birth Day) = 7/7 = 1
The probability for the second person to be born on the same day as the first person is P(Week Birth Day Second l Week Birth Day) = 1/7
By conditional probability, we have
P(B|A) = [tex]\frac{P(AnB}{P(A)}[/tex] that is [tex]\frac{1*\frac{1}{7} }{1} = \frac{1}{7}[/tex]
(b) Since there 7 days in a week, the probability that in a group of n people chosen at random, there are at least two born on the same day of the week where n ≥ 8 is 1 as there must be two people born on the same day
P(A) = 1 when n ≥ 8
The probability that at least two were born on the same day when n < 8 then there are
Number of ways to choose the birthday of first person = 7
Number of ways to choose the birthday of second person = 6
Number of ways to choose the birthday of third person =5
and so on till we have
Number of ways to choose the birthday of seventh person = 1
P(At least 2 born on the same day) can then be found by the complement rule thus
Let P(At least 2 born on the same day) = P(A)
Then we have P(A) = 1 - P(A)' where P(A)' = P (No 2 born on the same day)
Thus we have P(A) = 1 - [tex]\frac{7}{7} *\frac{6}{7}* \frac{5}{7} *\frac{4}{7} *\frac{3}{7}* \frac{2}{7} *\frac{1}{7}[/tex] = 1 - [tex]\frac{720}{117649}[/tex] = 0.99388
n = 6 we have P(A) = 1 - [tex]\frac{7}{7} *\frac{6}{7}* \frac{5}{7} *\frac{4}{7} *\frac{3}{7}* \frac{2}{7}[/tex] = 1 - [tex]\frac{720}{117649}/\frac{1}{7}[/tex] =1 - [tex]\frac{720}{16807}[/tex] =0.95716
n = 5 gives P(A) =1 - [tex]\frac{7}{7} *\frac{6}{7}* \frac{5}{7} *\frac{4}{7} *\frac{3}{7}[/tex] = 1 - [tex]\frac{720}{16807}/\frac{2}{7}[/tex] = 1- [tex]\frac{360}{2401}[/tex] = 0.850062
n = 4 P(A) =1 - [tex]\frac{7}{7} *\frac{6}{7}* \frac{5}{7} *\frac{4}{7}[/tex] = 1 - [tex]\frac{360}{2401} / \frac{3}{7}[/tex] = 1 - [tex]\frac{120}{343}[/tex] = [tex]\frac{223}{343}[/tex] = 0.65
n = 3 P(A) =1 - [tex]\frac{7}{7} *\frac{6}{7}* \frac{5}{7}[/tex] = 1 - [tex]\frac{120}{343}*\frac{7}{4}[/tex] = 1 -[tex]\frac{30}{49}[/tex] = [tex]\frac{19}{49}[/tex] = 0.38776
n = 2 P(A) =1 - [tex]\frac{7}{7} *\frac{6}{7}[/tex] = 1 -[tex]\frac{30}{49}*\frac{7}{5}[/tex] = [tex]\frac{1}{7}[/tex] = 0.14286
(c) For the probability to be greater than half that there are at least two people born on the same day of the week is given by
n = 4, P(A) = 0.65
n = 3, P(A) = 0.38776
Therefore for a probability greater than 4, n should be > 3
n > 3
