probability of a student being a sophomore is 20% because 20% of the population are sophomores.
probability of a student not being a sophomore is 80% because 80% of the population are not sophomores.
you sample 10 students.
the formula for exactly x students out of 10 being sophomores is uses the formula:
p(x) = .20^x * .80^(10-x) * 10Cx
The 10Cx is the combination formula for 10 things taken 2 at a time which is the possible ways that 2 things can be arranged out of 10 things assuming the 2 are identical to each other and the 8 are identical to each other but not identical to the 2.
your probabilities would be equal to:
x p(c)^x p(o)^(10-x) 10Cx p(x)
0 1 0.107374182 1 0.107374182
1 0.2 0.134217728 10 0.268435456
2 0.04 0.16777216 45 0.301989888 ***** most probable
3 0.008 0.2097152 120 0.201326592
4 0.0016 0.262144 210 0.088080384
5 0.00032 0.32768 252 0.026424115
6 0.000064 0.4096 210 0.005505024
7 0.0000128 0.512 120 0.000786432
8 0.00000256 0.64 45 7.3728E-05
9 0.000000512 0.8 10 0.000004096
10 1.024E-07 1 1 1.024E-07
sum of all probabilities >>>>>>>>>>>>>>>>>>> 1
the sum of all the probabilities is equal to 1 as it should be.
answer to a.
probability the sample contains exactly 2 is .301989888
answer to b.
probability the sample contains at least 3 sophomores is equal to the sum of (exactly 3 to exactly 10) which is also equal to 1 minus the sum of (exactly 0 to exactly 2).
that probability is equal to 0.322200474
answer to c.
this one's tougher to answer.
you can see from the probability table that .2 is the largest number in the table.
if this is a random sample and the samples are normally distributed, then the most probable value in each sample will be 2.
this is because the mean of the population is .2 which is the same as 2 out of 10.
if the probability of the number of sophomores in the population was .6, then the most probable value in the table would be 6 out of 10.
here's the same table with the probability of a sophomore being .6 and the probability of not a sophomore being .4
x p(c)^x p(o)^(10-x) 10Cx p(x)
0 1 0.000104858 1 0.000104858
1 0.6 0.000262144 10 0.001572864
2 0.36 0.00065536 45 0.010616832
3 0.216 0.0016384 120 0.042467328
4 0.1296 0.004096 210 0.111476736
5 0.07776 0.01024 252 0.200658125
6 0.046656 0.0256 210 0.250822656 ***** most probable
7 0.0279936 0.064 120 0.214990848
8 0.01679616 0.16 45 0.120932352
9 0.010077696 0.4 10 0.040310784
10 0.006046618 1 1 0.006046618
sum of all probabilities >>>>>>>>>>>>>> 1
you can see that the most probable value in each sample is now 6 out of 10 (p(x) = .250822656)
the most probable number of sophomores in each sample follows the probability of being a sophomore in the general popullation, whatever it is.
this assumes a normal distribution about the mean.
