Paying fg for help with basic, first year statistics such as binomial distribution, poisson distribution etc. Shoot me a pm

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1.

A race consists of 11 women and 8 men. (Give answer as a fraction or a decimal out to at least 4 places. If your answer is very small use scientific notation for example 3.3421E-6.)

What is the probability that the top three finishers were

A) All men

B). All women

C) 2 men and 1 woman

D) 1 man and 2 women


2.

A jury pool has 22 men and 21 women, from which 12 jurors will be selected. Assuming that each person is equally likely to be chosen and that the jury is selected at random, find the probability that the jury consists of the following.

A). All men

B). All women

C) 8 men and 4 women

D) 6 men and 6 women

3.

A) A Geiger counter counts the number of alpha particles from radioactive material. Over a long period of time, an average of 29 particles per minute occurs. Assume the arrival of particles at the counter follows a Poisson distribution.
Find the probability that at least one particle arrives in a particular one second period. Round your answer to four decimals.

B). Find the probability that at least two particles arrive in a particular 3 second period. Round your answer to four decimals.



This post was edited by Logarius on Feb 28 2018 08:32pm

1 and 2 are the same problem (assuming that every runner as an equal chance of winning the race or of finishing at any given place).

When you select A people among B, the number of choices is C(A,B) = B ! / (A ! x (B-A) !)

For example on 1 (question A) :
top three finishers are all men : C(3,8) = number of choices of selecting 3 people among 8 men.
C(3,8) = 8! / (3! x 5!) = 8x7x6x5x4x3x2x1 / ( 3x2x1 x 5x4x3x2x1 ) = 56
total number of possible finishers : C(3,19) = number of choices of selecting 3 people among 19 runners.
C(3,19) = 19! / (3! x 16!) = 19x18x17 / (3x2x1) = 969

Probabilty = 56 / 969 ~ 5.779 %

For example on 2 (question C) :
select 8 men among 22 : C(8,22) = 22x21x20x19x18x17x16x15 / (8x7x6x5x4x3x2x1) = 319770
select 4 women among 21 : C(4,21) = 21x20x19x18 / (4x3x2x1) = 5985
total number of jurys with 8 men and 4 women = 319770 x 5985 = 1913823450
total number of jurys : select 12 people among 43 : C(12,43) = ... = 15338678264

Probability = 1913823450 / 15338678264 ~ 12.477 %

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For a Poisson distribution, probability of being equal to k is : λ^k . exp(-λ) / k!
where λ is the average value.

A) 'at least 1 arrives' is the opposite of 'none arrives'.
Here, λ = 29/60 per second (average 29 particles over 60 seconds).
Probability of none arriving is : λ^0 . exp(-λ) / 0! = exp(-λ)
Probability of at least 1 arriving is : 1 - exp(-λ) ~ 38.3276 %

B ) 'at least 2 arrives' is the opposite of 'none or only 1 arrives'.
Now, λ = 29/20 per 3 seconds (average 29 particles over 20x3 seconds).
Probability of none or 1 arriving is : λ^0 . exp(-λ) / 0! + λ^1 . exp(-λ) / 1!= (1+λ).exp(-λ) ~ 57.4697 %
Probability of at least 2 arriving : ~ 100% - 57.4697% = 42.5303 %

Feel free to pm if something is not clear.
Good luck !

This post was edited by feanur on Feb 28 2018 09:38pm