Question

In a randomized trial, 200 subjects were to be randomly assigned either to the drug treatment group or to the placebo group. Researchers flipped a coin to decide each participant’s treatment assignment. After randomization, 112 subjects were in the treatment group and 88 were in the control group. What’s the probability that 112 or more subjects could end up in the treatment group purely by chance? Hint: use a normal approximation to the binomial.

Answer 1

let`s start
tell what is the probability of exactly 112 subjects getting assigned to treatment group ?

Answer 2

50%

Answer 3

tell me the method(explanation) you used

Answer 4

flipping a coin.

Answer 5

well you answer is not correct, let`s go step by step

we have to decide for 1st subject, so we flip the coin
what are the chances that this subject lands in the treatment side ?

Answer 6

when we flip a coin 1/2 chances he would land in treatment side 1/2 chances he would land in other

Answer 7

great
now take just the second subject and we flip a coin to decide,
what are the chances that this subject lands in the treatment side ?

Answer 8

1/2

Answer 9

great now if i have to consider the events simultaneously i.e i have 2 subjects and the destiny of the subjects will be decided by the flip of the coin
what are the chances that both of these fall on the treatment side ?

Answer 10

chances of 1 plus chances of 2

Answer 11

no
plus is used when they are complementary events like the occurrence of 1st subject on the treatment side rules out the occurrence of 2nd subject on the treatment side

multiplication is used when both the sub events are part of the major event, or they are independent of each other
like in our case 1st subject going into treatment side has not effect on the probability of the 2nd subject

so now tell me the answer again?

Answer 12

ok it would be 1/2 x 1/2 or .5 x .5

Answer 13

yes
now tell me what is the probability of getting exactly 112 subjects on treatment side?

Answer 14

112 x .5?

Answer 15

no
(0.5)*(0.5)*(0.5) so on 112 times (THINK WHY??)
i.e (1/2)^112

where * = multiply
^ = raised to power
tell me when this step is clear as this is very important

Answer 16

yes got that.clear

Answer 17

ok now tell me the probability for exactly 113 such cases ?

Answer 18

.5x.5x.5............113 times

Answer 19

yes
now you should realize that occurrence of exactly 112 and exactly 113 is complementary to each other
clear ?

Answer 20

yes

Answer 21

so probability of 112 or more landing on treatment side is
(1/2)^112 + (1/2)^113 + ... +(1/2)^200
clear ?

Answer 22

yes.clear till here

Answer 23

so all you need now is a way to find the sum
it`s a GP which ratio 2

Answer 24

how to make now these big calculations and find a whole number percentage

Answer 25

check this out
http://en.wikipedia.org/wiki/Geometric_progression

Answer 26

confused :(

Answer 27

S = a + ar +ar^2 + ar^3 + ...+ar^(n-1)
S = a(1-r^n)/(1-r)
a = (1/2)^112 and r = 1/2

Answer 28

what is n?

Answer 29

you can find out
ar^(n-1) = last term = (1/2)^200
a and r = you know
so find out n