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maria [59]
3 years ago
13

How many randomly chosen guests should I invite to my party so that the probability of having a guest with the same birthday as

mine is at least 2/3?
Mathematics
1 answer:
dimulka [17.4K]3 years ago
5 0

Answer:

At least 401 chosen guests

Step-by-step explanation:

Represent those with same birth day with Same

So:

P(Same) > \frac{2}{3}

Represent number of people with n

There are 365 days in a year and 364 days out of these days is not your birthday.

So, there the probability that n people do not share your birthday is (\frac{364}{365})^n

i.e.

P(none) = (\frac{364}{365})^n

Solving further, we have that:

P(same) + P(none) = 1

P(same) + (\frac{364}{365})^n = 1

P(same) = 1 - (\frac{364}{365})^n

We're calculating the probability that P(Same) > \frac{2}{3}.

So, we have:

1 - (\frac{364}{365})^n > \frac{2}{3}

Collect Like Terms

1 - \frac{2}{3}> (\frac{364}{365})^n

\frac{3-2}{3}> (\frac{364}{365})^n

\frac{1}{3}> (\frac{364}{365})^n

3^{-1} > (\frac{364}{365})^n

Take natural logarithm (ln) of both sides

ln(3^{-1}) > ln((\frac{364}{365})^n)

-ln\ 3 > n\ ln\ (\frac{364}{365})

Apply laws of logarithm

-ln\ 3 > n\ (ln(364) - ln(365))

Multiply through by -1

ln\ 3 < n\ (ln(365) - ln(364))

Solve for n

\frac{ln\ 3}{(ln(365) - ln(364))} < n

Reorder

n > \frac{ln\ 3}{(ln(365) - ln(364))}

n > \frac{1.09861228867}{(5.89989735358 - 5.89715386764)}

n> \frac{1.09861228867}{0.00274348594}

n > 400.443928891

n > 400 (approximated)

This implies that n = 401, 402, 403 .....

i.e

n \geq 401

So, at least 401 people has to be invited

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