Mathematics, 10.10.2019 04:00 BlackMilkTea

Consider the experiment, called the birthday problem , where our task is to determine the probability that in a group of people of a certain size there are at least two people who have the same birthday (the same month and day of month). suppose there is a room with 9 people in it, find the probability that at least two people have the same birthday. ignore leap years; assume each year has 365 days.

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Answer from: mimivenga

There is a 0.02% probability that at least two people have the same birthday.

Step-by-step explanation:

There are only two possible outcomes: Either the people do have the same birthday or they do not. So we use the binomial probability distribution.

Binomial probability

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.\pi^{x}.(1-\pi)^{n-x}$

In which $C_{n,x}$ is the number of different combinatios of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And $\pi$ is the probability of X happening.

In this problem, we have that:

There are 365 days in a year, so the probability that a person has a birthdday on any given day is given by $\pi = \frac{1}{365} = 0.0027$ .

Suppose there is a room with 9 people in it, find the probability that at least two people have the same birthday.

There are 9 people, so n = 9 .

We also want to find P(X 1)

And

$P(X \leq 1) + P(X 1) = 1$

$P(X 1) = 1 - P(X \leq 1)$

We also have that:

$P(X \leq 1) = P(X = 0) + P(X = 1)$

$P(X = x) = C_{n,x}.\pi^{x}.(1-\pi)^{n-x}$

$P(X = 0) = C_{9,0}.(0.0027)^{0}.(0.9973)^{9} = 0.9760$

$P(X = 1) = C_{9,1}.(0.0027)^{1}.(0.9973)^{8} = 0.0238$

--------

$P(X \leq 1) = P(X = 0) + P(X = 1) = 0.9760 + 0.0238 = 0.9998$

$P(X 1) = 1 - P(X \leq 1) = 1 - 0.9998 = 0.0002$

There is a 0.02% probability that at least two people have the same birthday.

Answer from: Quest

g(x) is the negative form of f(x)? because whenever its the negative form it reflects its self over the x axis. idk if this is right though.

step-by-step explanation:

Answer from: Quest

the correct functions are

h(x) = 3x + 2

g(x) = 2^x

step-by-step explanation:

i proceed to calculate the different values of k (x)

case 1) k(x) = g(x) ∘ h(x)

g(x) ∘ h(x)=g(h(x))

k(x) = [2]^[3x + > is the option 4. k(x) = 2^(3x + 2 )

case 2) k(x) = g(x) + h(x)

k(x) = [2^x]+[3x + 2]=2^x+3x+> is the option 1.) k(x) = 2^x + 3x + 2

case 3)k(x) = h(x) ÷ g(x)

k(x) = [3x + 2]/[2^x]=3x/(2^x)+2/(2^x)=3x*(2^-x)+2^(1-x)

is the option 3.) k(x) = (3x)2^(-x) + 2^(-x + 1 )

case 4)k(x) = g(x) - h(x)

k(x) = [2^x]-[3x + 2]=2^x-3x-> is the option 2.) k(x) = 2^x - 3x - 2

case 5) k(x) = g(x) × h(x)

k(x) = [2^x]*[3x + > is the option 5.) k(x) = 2^x(3x + 2)

case 6)k(x) = h(x) ∘ g(x)

h(x) ∘ g(x)=h(g(x))

k(x)=3*[2^x]+> is the option 6. k(x) = 3(2^x) +2

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Consider the experiment, called the birthday problem , where our task is to determine the probabilit...