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What is the probability that in a room of 8 people 2 have the same birthday?
P = 0.072314699... ≈ 7.23%SOLUTIONLet's identify the 8 people by integers 1, 2,..., 8, and their birthday by x1, x2,..., x8.To include the possibility of xi = Feb. 29 (of a leap year), we will consider 4(365) +1 = 1451 days, so the probability of a person having Feb. 29 as his birthday isP(xi = Feb. 29) = 1/1461. The probability of a person having his birthday on anygiven day except Feb. 29 is P(xi ≠ Feb. 29) = 4/1461.Let's first calculate the probability of person 1 and 2 having the same birthday, x1 = x2, and persons 3, 4,..., 8, all have different birthdays x3,...,x8.Now we go.We take person 1 and have two possibilities:A).- x1 = Feb. 29 B).- x1 ≠ Feb. 29P(x1 = Feb. 29) = 1/1461; P(x ≠ Feb. 29) = 1 - 1/1461We will construct two branches. One for each possibility.Branch A: given x1 = Feb. 29, the probability of person 2 having the same birthdayis P(x2 = x1) = 1 - 1/1461. The probability of person 3 not having the same birthday is P(x3 ≠ x1 = x2) = 1 - 1/1461. The probability of person 4 not having either two birthdays is P(x4 ≠ x1, x4 ≠ x3) = 1 - 5/1461. The probability of person 5 not sharing x1 nor x3 nor x4 is P(x5 ≠ x4, x5 ≠ x3, x5 ≠ x2) = 1 - 9/1461. And so a series of terms develops:P(xi ≠ ... ) = 1 - (4i - 3)/1461 from i = 1, to i = n-2where n is the number of people in the room (In this particular question for 8people, n = 8).The branch we have constructed for possibility A is the product of the individualconditional probabilities;A: (1/1461)2 π1n-2[1-(4i-3)/1461]where π1n-2 is the product operator "pi" of "i" terms from i =1, to i = n-2.Branch B: Given x1 ≠Feb. 29, the probability of person 2 having the same birthdayis P(x2 = x2) = 4/1461. The probability of person 3 not having the same birthdayis P(x3 ≠ x1) = 1 - 4/1461. The probability of person 4 not sharing birthdays with 2 and 3 is P(x4 ≠ x3, x4 ≠ x2) = 1 - 8/1451. The probability of person 5 notsharing x2 nor x3 nor x4 is P(x5 ≠ ...) = 1 - 12/1461. And so a series of termsdevelops:P(xi ≠ ... ) = 1- (4i -4)/1461 from i = 2, to i = n-1where n is the number of people in the room (In this particular question for 8 people, n = 8).Branch for possibility B: (1-1/1461)(4/1461) π2n-1[1-(4i-4)/1461]The sum of these two branch expressions give the probability of persons 1 and 2sharing a birthday in a group of n persons:P(x1=x2)=(1/1451)2π1n-2[1-(4i-3)/1461]+(1-1/1461)(4/1461)π2n-1[1-(4i-4)/1461]Now, the probability of any two persons, and only those two persons, to share a birthday in a group of n persons, is the previous probability, times the number of different combinations of two persons, (i,j), in a group of n persons, nC2(where, nC2 = n!/[2!(n-2)!] )The final expression is:--------------------------------------------------------------------------------------------P = nC2{(1/1451)2π1n-2[1-(4i-3)/1461]+(1-1/1461)(4/1461)π2n-1[1-(4i-4)/1461]}--------------------------------------------------------------------------------------------For the case of n = 8, the result is, P = 0.072314699... ≈ 7.23%
If Two dice are rolled Find the probability of getting a sum greater than 8?
The first die can be from 2 to 6, it can not be one. So, the first die has a 5/6 chance. The second die must be a 2 if 6 is on the first die, 3 if 5 is on the first die, etc. so it has a 1/6 chance. Since the two events must occur ("and" situation) and they are independent: (1/6) * 5/6 = 5/36. A second way of thinking about this is, that each die can have values from 1 to 6, so you can make a x-y table (1-6 columns and 1-6 rows) and see 36 combinations, of which only 5 will add to 8: (2,6) (6,2) (3,5) (5,3) (4,4) so the answer is 5/36. We know that there are 36 different possible dice rolls. Since the dice must be greater than 8, we have the following combinations: 3-6 x2 4-5 x2 4-6 x2 5-5 5-6 x2 6-6 Which is 10, or 5/18. If you looking for > or = to 8, we must add: 2-6 x2 3-5 x2 4-4 to this list. Which is 15 => 5/12
How can you use the digit 8 four times to make 89?
88 + 8/888 + 8/8 = 89
What is the probability of getting one head when flipping a coin three times?
The probability is 3/8.The probability is 3/8.The probability is 3/8.The probability is 3/8.
What is the answer of the median if it is between 8 and 8?
well if by between 8 and 8 you mean 8+8 then divide it by two to get......8.- Yep, you are absolutely right!! if there were 3 numbers in a data set then u would have to cross keep crossing out numbers from each side until you get a remainder of only one number. In some cases you will not get ONLY one number. So, you would have to:1) Add 8 + 82) You get 16. So divide 16 by two because there are only two numbers in the set.3) You get 8.4) The median of 8 and 8 is 8!
