In a large university, 15% of the students are female. If a random sample of twe
ID: 2922768 • Letter: I
Question
In a large university, 15% of the students are female. If a random sample of twenty students is selected a. what is the probability that the sample contains exactly four female students? b. what is the probability that the sample will contain no female students? c. what is the probability that the sample will contain exactly twenty female students? d. what is the probability that the sample will contain more than nine female students? e. what is the probability that the sample will contain fewer than five female students? f. what is the expected number of female students?Explanation / Answer
Probability that the student is female = p = 15/100 = 0.15 =
q = 1-p = 1-0.15 = 0.85
Sample size n= 20
This is a binomial distribution problem
Probability that r students out of twenty selected are female is given by = nCr * p^r * q^(n-r)
a) Probability that exactly 4 female students are there in the sample
P(r=4) = 20C4 * (0.15^4) * (0.85^16)
= 0.1821
b) sample has no female student
P(r = 0) = 20C0 * (0.15^0) * ( 0.85^20)
= 0.0387
c) sample contain exactly twenty female student
P(r=20) = 20C20 * (0.15^20) * (0.85^0)
= 3.325 * 10^-17
Using the above formula we will calculate the probability for each value of r to make our solution easy
d) More than nine students are female
= P(10) +P(11)+ P(12) +...................+P(20)
from the above table we can add the required probability
= 0.000248
e) Fewer tan five students
= P(4) + P(3) + P(2) + P(1) + P(0)
= 0.1821 + 0.2428 + 0.2293 + 0.1367 + 0.0387
= 0.8298
f) Expected number of female student
= summation (r.P(r))
= 0*P(0) + 1*P(1) + 2*P(2) + 3*P(3) +.................+ 19*P(19) + 20*P(20) = 3
from the table we can calculate it to be 3
r nCr p^r q^(n-r) P(r) r.P(r) 0 1 1 0.038759531 0.038759531 0 1 20 0.15 0.045599448 0.136798345 0.136798 2 190 0.0225 0.05364641 0.229338402 0.458677 3 1140 0.003375 0.063113423 0.242828896 0.728487 4 4845 0.00050625 0.074251086 0.182121672 0.728487 5 15504 7.59375E-05 0.087354219 0.10284518 0.514226 6 38760 1.13906E-05 0.10276967 0.045372873 0.272237 7 77520 1.70859E-06 0.120905494 0.016013955 0.112098 8 125970 2.56289E-07 0.142241757 0.004592237 0.036738 9 167960 3.84434E-08 0.167343244 0.001080526 0.009725 10 184756 5.7665E-09 0.196874404 0.000209749 0.002097 11 167960 8.64976E-10 0.231616946 3.36496E-05 0.00037 12 125970 1.29746E-10 0.272490525 4.45362E-06 5.34E-05 13 77520 1.9462E-11 0.320577088 4.83652E-07 6.29E-06 14 38760 2.91929E-12 0.377149516 4.26751E-08 5.97E-07 15 15504 4.37894E-13 0.443705313 3.01236E-09 4.52E-08 16 4845 6.56841E-14 0.52200625 1.66123E-10 2.66E-09 17 1140 9.85261E-15 0.614125 6.89784E-12 1.17E-10 18 180 1.47789E-15 0.7225 1.922E-13 3.46E-12 19 20 2.21684E-16 0.85 3.76862E-15 7.16E-14 20 1 3.32526E-17 1 3.32526E-17 6.65E-16 Total 1 3Related Questions
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