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Question
A computerized dating service has a list of 50 men and 50 women. Names are selected at random; how many names must be chosen to guarantee one name of each gender?
There could be (with a very small chance), if 50 are drawn, that all 50 aare of the same sex.
To guarantee it, it would need to be 51.
The probability of all 50 being of all boys is 1/2^50, which is around 9x10E-16.
As can be seen, this is small. This needs to be doubled, for we could also select 50 girls.
Thus, it is 1.8x10E-15.
The probability of 49 being selected that were all boys would be 50/2^50.
For all girls, it would be the same. 50+50=100, so it would be 100/2^50,
which is around 9x10E-14.
The probability of 48 being selected that were all boys would be 1225/2^50,
so both would be 2250/2^50, which is around 2x10E-12.
The probability of 47 being selected that were all boys would be ...
well, I think I'll give a table.
Column 1 is the number selected.
Column 2 is the number of ways of all boys or all girls.
Column 3 is the chance of them all being the same sex.
As can be seen, as long as 30 or more are selected,
there is over a 90% chance of at least one of each sex being drawn
since the probability of them all being one sex is less than 10%.
It is also to be noted that to get better than 0.00000001, more than 45 should be selected.
50 2 1.77636E-15
49 100 8.88178E-14
48 2450 2.17604E-12
47 39200 3.48166E-11
46 460600 4.09095E-10
45 4237520 3.76367E-09
44 31781400 0.000000028
43 199768800 0.000000177
42 1073757300 0.000000954
41 5010867400 0.000004451
40 20544556340 0.000018247
39 74707477600 0.000066354
38 2.42799E+11 0.000215649
37 7.09721E+11 0.000630359
36 1.87569E+12 0.001665949
35 4.50166E+12 0.003998277
34 9.84738E+12 0.008746230
33 1.96948E+13 0.017492460
32 3.61071E+13 0.032069510
31 6.08119E+13 0.054011805
30 9.42584E+13 0.083718299
29 1.34655E+14 0.119597569
This is probably a lot more than was needed, but it is so much fun for me to do!
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Answers by Expert:
Scott A Wilson
Expertise
I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can even tell you it takes me over 2,000 steps to go a mile, but is that relevant?
Experience
Experience in the area; I have tutored people in the above areas of mathematics for almost two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree almost 25 years ago. In just two more years, I received an MS degree as well, but more on that later.
I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman.
I tutored at Mathnasium for well over a year.
I worked at The Boeing Company for over 5 years.
I received an MS degreee in Mathematics from Oregon State Univeristy.
The classes I took were over 100 hours of upper division credits in mathematical courses such as
calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more.
I graduated with honors in both my BS and MS degrees.
Past/Present Clients: College Students at Oregon State University, various math people since college,
over 7,500 people on the PC from the US and rest the world.
Publications
My master's paper was published in the OSU journal.
The subject of it was Numerical Analysis used in shock waves and rarefaction fans.
It dealt with discontinuities that arose over time.
They were solved using the Leap Frog method.
That method was used and improvements of it were shown.
The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.
Education/Credentials
Master of Science at OSU with high honors in mathematics.
Bachelor of Science at OSU with high honors in mathematical sciences.
This degree involved mathematics, statistics, and computer science.
I also took sophmore level physics and chemistry while I was attending college.
On the side I took raquetball, but that's still not relevant.
Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State.
I was in near the top in most of my classes. In several classes in mathematics, I was first.
In a class of over 100 students, I was always one of the first ones to complete the test.
I graduated with well over 50 credits in upper division mathematics.
Past/Present Clients
My clients have been students at OSU, people nearby, friends with math questions,
and several people every day on the PC, and you're probably make one more.
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