1.
MSV 11: Binomial Reverse

2. What is the most likely value for X?
Question 1: X ~ B(11, 0.7).
What is the most likely value for X?
What happens if we turn this question around?
Question 2: X ~ B(11, p)
and 8 is the most likely value for X.
What range of values can p take?

3. Binomial Reverse: Answers
Question 1. The mean of the distribution is 7.7,
so the only candidates for most likely value are 7 and 8.
(The fact that 7.7 is nearer to 8 than 7 does not automatically guarantee that 8 is going to be the answer.)
11C7(0.7)7(0.3)4 = 0.220
11C8(0.7)8(0.3)3 = 0.257
So 8 is the most likely value.

4. 11C7p7q4 = 11C8p8q3 , Question 2. The two boundary situations are:
Taking the left-hand picture, we need to solve:
11C7p7q4 = 11C8p8q3 ,
which gives 330q = 165p.

5. 11C8p8q3 = 11C9p9q2 , Taking the right-hand picture, we need to solve:
which gives 165q = 55p.

6. is written by Jonny Griffiths
is written by Jonny Griffiths